G be a graph, and the number of components of G is denoted by c(G). Let t be a positive real number. A connected graph G is t-tough if tc(G − S) ≤ |S| for every vertex cut S of V(G). The toughness of G is the largest value of t for which G is t-tough, denoted by τ(G). We call a graph G Hamiltonian if it has a cycle that contains all vertices of G. Chvátal and other scholars investigate the relationship between toughness conditions and the existence of cyclic structures. In this paper, we establish some sufficient conditions that a graph with toughness is Hamiltonian based on the number of edges, spectral radius, and signless Laplacian spectral radius of the graph.Let MR subject classifications: 05C50, 15A18. G =  be a finite simple undirected graph with vertex set V(G) = {v1, v2, …, vn} and edge set E(G). Write by m = |E(G)| the number of edges and n = |V(G)| the number of vertices of the graph G, respectively. The set of neighbors of a vertex v in graph G is denoted by NG(v). Let vi ∈ V(G), we denote by di = dvi = dG(vi) = |NG(vi)| the degree of vi. Denote by δ(G) [Δ(G)] or simply δ (Δ) the minimum (maximum) degree of G. Let  be a nondecreasing degree sequence of G, that is, d1 ≤ d2 ≤ ⋯ ≤ dn. For convenience, we use G, where xk is the number of vertices of degree k in the graph G. We denote a bipartite graph with bipartition  by using G. We denote the cycle and the complete graph on n vertices by using Cn and Kn, respectively. We use Km, n to denote a complete bipartite graph with two parts having m, n vertices, respectively. Let G and H be two disjoint graphs. We denote by G+H the disjoint union of G and H, which is a graph with vertex set V(G) ∪ V(H) and edge set E(G) ∪ E(H). If G1 = G2 = … = Gk, we denote G1 + G2 + ⋯ + Gk by kG1. We denote by G ∨ H the join of G and H, which is a graph obtained from the disjoint union of G and H by adding edges joining every vertex of G to every vertex of H. Let Kn − 1 + v denote the complete graph Kn − 1 together with an isolated vertex v. Other undefined symbols reference can be seen in Bondy and Murty (Let nd Murty  and Bauend Murty .adjacency matrix of G is A(G) = (aij), where aij = 1 if vi and vj are adjacent in G and aij = 0 otherwise. Let D(G) be the degree diagonal matrix of G, i.e., D(G) = diag{dG(v1), dG(v2), …, dG(vn)}. The matrix Q(G) = D(G) + A(G) is called the signless Laplacian matrix of G. The largest eigenvalue of A(G), denoted by μ(G), is called to be the spectral radius of G. The largest eigenvalue of Q(G), denoted by q(G), is called to be the signless Laplacian spectral radius of G.The G is called a Hamilton graph if it has a Hamilton cycle, and then we also call G Hamiltonian. The number of components of G is denoted by c(G). Let t be a positive real number. A connected graph G is t-tough if tc(G − S) ≤ |S| for every vertex cut S of V(G). The toughness of G is the largest value of t for which G is t-tough, denoted by τ(G). If G is a complete graph, take τ(Kn) = ∞ for all n ≥ 1. If G is not a complete graph, G. Obviously, a t-tough graph is s-tough for all s < t.A cycle (path) containing every vertex of a graph is called a Hamilton cycle (path) of the graph. Graph NP-complete problem. In recent years, the study of Hamiltonian problem using spectrum graph theory has received extensive attention, and some meaningful results are obtained, such as Fiedler and Nikiforov  of the graph G, if there is a one-to-one mapping φ from vertex set V(G) of the graph to the components of the vector X; simply written Xu = φ(u).At the beginning of this section, we first give some definitions. Let A(G) corresponding to the eigenvector X if and only if X ≠ 0,When μ is an eigenvalue of the adjacency matrix characteristic equation of G.The Equation (2.1) is called the q is an eigenvalue of signless Laplacian matrix Q(G) corresponding to the eigenvector X if and only if X ≠ 0,When signless Laplacian characteristic equation of G.The Equation (2.2) is called the Lemma 2.1. Hoàng , and G is t-tough, then δ(G) ≥ 2t.Proof Let δ(G) = dv, S = NG(v), then |S| = |NG(v)| = δ(G). We can get c(G − S) ≥ 2. Because G is not a complete graph, bythenG) ≥ 2t.thus, we can get δ(The proof is completed.■Lemma 2.6. Jung (G be a graph without a Hamiltonian cycle and at least 11 vertices. Then.6. Jung  let G bei) there exist two non-adjacent vertices x, y such that d(x) + d(y) ≤ |V(G)| − 5 or(ii) there exist for some t ≥ 1 vertices x1, x2, …, xt such that G − x1 − ⋯ − xt has at least t + 1 components. + d(y) ≤ |V(G)| − 5.Proof Because G has no Hamiltonian cycle, G is not a complete graph. If there exist for some s ≥ 1 vertices x1, x2, …, xs such that G − x1 − ⋯ −xs has at least s + 1 components, bythena contradiction.The proof is completed.■G, denoting by G by recursively joining pairs of nonadjacent vertices whose degree sum is at least n until no such pair remains, refer to Bondy and Chvátal  and simple connected graph with n(≥ 8t) vertices and m edges. IfthenG is Hamiltonian when t ∈ {1, 2} and n ≥ 8t.(i) G is Hamiltonian when t = 3 and n > 9t.(ii) Proof Suppose that G is not a Hamilton graph. By Lemma 2.1, there exists a positive integer k for dk ≤ k, such that dn − k + t ≤ n − k − 1. Then we havethusk − 2t)(2n − 3k − 5t − 1) ≤ 0. Next, we discuss three cases.Since Case 1t = 1.n ≥ 8t = 8, (k − 2)(2n − 3k − 6) ≤ 0. By Lemma 2.5, δ (G)≥ 2t = 2. Since δ(G) ≤ dk ≤ k, then k ≥ 2.In this case, Case 1.1 (k − 2)(2n − 3k − 6) = 0, i.e., k = 2; or k ≠ 2 and 2n − 3k − 6 = 0.Case 1.1.1k = 2.G is a graph with d2 ≤ 2, dn − 1 ≤ n−3, dn ≤ n − 1, and we have G must be with degree sequence .In this case, G are non-adjacent, G must be the graph K1 ∨ (Kn − 3 − uv + wu + zv), where u, v ∈ V(Kn − 3), w, z ∉ V(Kn − 3), is Hamiltonian, a contraction.If two 2-degree vertices of G are adjacent, G must be the graph K1 ∨ (K2 + Kn − 3), but If two 2-degree vertices of Case 1.1.2k ≠ 2 and 2n − 3k − 6 = 0.n ≤ 11 because n = 9, k = 4. Then d4 ≤ 4, d6 ≤ 4, d9 ≤ 8, and we have G must be with degree sequence , and G = K3 ∨ (3K2) is Hamiltonian, a contradiction.In this case, we can get Case 1.2 (k − 2)(2n − 3k − 6) < 0.k ≥ 3 and 2n − 3k − 6 < 0. Since n ≥ 2k + 1. From these results, we have 4k + 2 ≤ 2n ≤ 3k + 5, that is k ≤ 3. Thus, we have k = 3, n = 7. A contradiction with known condition n ≥ 8.In this case, Case 2t = 2.k − 4)(2n − 3k − 11) ≤ 0. By Lemma 2.5, δ(G)≥ 2t = 4. Since δ(G) ≤ dk ≤ k, then k ≥ 4.In this case, (Case 2.1 (k − 4)(2n − 3k − 11) = 0, i.e., k = 4; or k ≠ 4 and 2n − 3k − 11 = 0.Case 2.1.1k = 4.G is a graph with d4 ≤ 4, dn − 2 ≤ n − 5, dn ≤ n − 1, and we have G is the graph with degree sequence n − 6, (n − 1)2), and n ≥ 8t = 16 for t = 2. Let S is the set containing four 4 − degree vertices of G, and there exist two non-adjacent vertices in S by Corollary 2.7.In this case, Lemma 2.8, S form a complete graph Kn − 4 in Kn − 4 is adjacent to one vertex of S, it must be adjacent to all vertices of S. Moreover, there are at least 2 vertices of the Kn − 4 adjacent to all vertices of S because there exist two non-adjacent vertices in S by Corollary 2.7.By Kn − 4 adjacent to all vertices of S, then K2 ∨  is Hamiltonian, then If there are 2 vertices of the Kn − 4 adjacent to all vertices of S, then K3 ∨ (Kn − 7 + 2K2) is Hamiltonian, then If there are 3 vertices of the Kn − 4 adjacent to all vertices of S, then K4 ∨ (Kn − 8 + 4K1) is Hamiltonian, then If there are 4 vertices of the Kn − 4 adjacent to all vertices of S, Ki ∨ (Kn − 4 − i + 4K1) is Hamiltonian, then If there are more than 4 vertices of the Case 2.1.2k ≠ 4 and 2n − 3k − 11 = 0.n ≤ 21 because t ≤ n, hence n = 16, k = 7. Then d7 ≤ 7, d11 ≤ 8, d16 ≤ 15, and we have 7, 84, 155). Then there are no two vertices x, y that are not adjacent such that d(x) + d(y) ≤∣ V(G) ∣ − 5. By Corollary 2.7, we can get G is Hamiltonian, a contradiction.In this case, we can get 16 ≤ Case 2.2 (k − 4)(2n − 3k − 11) < 0.k ≥ 5 and 2n − 3k − 11 < 0. Since n ≥ 2k + 1. From these results, we have 4k + 2 ≤ 2n ≤ 3k + 10, that is k ≤ 8. Thus, we have 16 = 8t ≤ n ≤ 17.In this case, n = 16, we have k = 7 from 4k + 2 ≤ 2n ≤ 3k + 10. At this time 2n − 3k − 11 = 0, which contradicts to 2n − 3k − 11 < 0.When n = 17, we have k = 8 from 4k + 2 ≤ 2n ≤ 3k + 10. In this case, d8 ≤ 8, d11 ≤ 8, d17 ≤ 16. We have When Corollary 2.7, we can get G is Hamiltonian, a contradiction.When When From 10,166) and . They are not graphic, a contradiction.(1) for degree sequence . If the corresponding graphs are not Hamiltonian, there must exist two non-adjacent 6-degree vertices by Corollary 2.7, and the corresponding graphs are isomorphic to K6 ∨ (C9 + 2K1) or K6 ∨ (C4 + C5 + 2K1) or K6 ∨ (C6 + K3 + 2K1). We can find these graphs are Hamiltonian, a contraction.(2) for degree sequence , , and  in x, y that are not adjacent such that d(x) + d(y) ≤ ∣V(G)∣ − 5. By Corollary 2.7, we can get that G is Hamiltonian, a contradiction.(3) the other degree sequence except (2n − 3k − 16) ≤ 0. By Lemma 2.5, δ(G) ≥ 2t = 6. Since δ ≤ dk ≤ k, then k ≥ 6.In this case, (Case 3.1 (k − 6)(2n − 3k − 16) = 0, i.e., k − 6 = 0 or k ≠ 6 and 2n − 3k − 16 = 0.Case 3.1.1k = 6.G is a graph with d6 ≤ 6, dn − 3 ≤ n − 7, dn ≤ n − 1. We have G is n − 9, (n − 1)3), and we have n > 9t = 27 because t = 3. Let S is the set containing six 6 − degree vertices of G.In this case, Lemma 2.8, S form a complete graph Kn − 6 in Kn − 6 is adjacent to one vertex of S, it must be adjacent to all vertices of S. Moreover, there are at least 2 vertices of the Kn − 6 adjacent to all vertices of S because there exist two non-adjacent vertices in S by Corollary 2.7.By Kn − 6 adjacent to all vertices of S, then K2 ∨ (Kn − 8 + C6) is Hamiltonian, then If there are 2 vertices of the Kn − 6 adjacent to all vertices of S, then K3 ∨ (Kn − 9 + C6) is Hamiltonian, then If there are 3 vertices of the Kn − 6 adjacent to all vertices of S, then K4 ∨ (Kn − 10 + C6) and K4 ∨ (Kn − 10 + 2C3) are Hamiltonian, then If there are 4 vertices of the Kn − 6 adjacent to all vertices of S, then K5 ∨ (Kn − 11 + 3K2) is Hamiltonian, then If there are 5 vertices of the Kn − 6 adjacent to all vertices of S, then Cn(G) is Hamiltonian, a contradiction.If there are 6 vertices of the Kn − 6 adjacent to all vertices of S, Ki ∨ (Kn − 6 − i + 6K1)(i > 6) is Hamiltonian, then If there are more than 6 vertices of the Case 3.1.2k ≠ 6 and 2n − 3k − 16 = 0.n ≤ 31 because n > 9t = 27. Since 2n − 3k − 16 = 0, then n = 29, k = 14. So d14 ≤ 14, d18 ≤ 14, d29 ≤ 28. We have 18, 2811). For this degree sequence, we can get that there are no two vertices x, y that are not adjacent such that d(x) + d(y) ≤ ∣V(G)∣ − 5. By Corollary 2.7, we can get G is Hamiltonian, a contradiction.In this case, we can get 28 ≤ Case 3.2 (k − 6)(2n − 3k − 16) < 0.k ≥ 7 and 2n − 3k − 16 < 0. Since n ≥ 2k + 1. From these results, we have 4k + 2 ≤ 2n ≤ 3k + 15, that is k ≤ 13. Thus, we have n ≤ 27. A contradiction with known condition n > 9t = 27. ■In this case, Theorem 3.2 Let G be a t − tough and simple connected graph with n vertices and m edges. IfthenG is Hamiltonian when t ∈ {1, 2} and n ≥ 8t.(i) G is Hamiltonian when t = 3 and n > 9t.(ii) Proof Suppose that G is not Hamiltonian. Since Kn is Hamiltonian, then G ≠ Kn. By theorem conditions, we can get n ≥ 8 when t ∈ {1, 2, 3}, thus G ≠ Kn − 11, .Lemma 2.2,By thenTheorem 3.1, we get G is Hamiltonian, a contradiction. ■By Theorem 3.3 Let G be a t − tough and simple connected graph with n vertices and m edges. IfthenG is Hamiltonian when t ∈ {1, 2} and n ≥ 8t.(i) G is Hamiltonian when t = 3 and n > 9t.(ii) Proof Suppose that G is not a Hamilton graph. Since Kn is Hamiltonian, then G ≠ Kn and G ≠ Kn − 1 + v. By theorem conditions, we can get n ≥ 8 when t ∈ {1, 2, 3}, thus G ≠ Kn, G ≠ Kn − 11, .Lemma 2.3,By thenTheorem 3.1, we get G is Hamiltonian, a contradiction.By ■We suggest the following general problems.Problem 1 Let G be a t − tough and simple connected graph with n vertices and m edges. Ifthent ∈ {1, 2} and n ≥ 8t, G is pancyclic.(i) when t = 3 and n > 9t, G is pancyclic.(ii) when Problem 2 Let G be a t − tough and simple connected graph with n vertices and m edges. Ifthent ∈ {1, 2} and n ≥ 8t, G is pancyclic.(i) when t = 3 and n > 9t, G is pancyclic.(ii) when Problem 3 Let G be a t − tough and simple connected graph with n vertices and m edges. Ifthent ∈ {1, 2} and n ≥ 8t, G is pancyclic.(i) when t = 3 and n > 9t, G is pancyclic.(ii) when The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.GY provide writing ideas and master the thesis as a whole. GC contributed significantly to analysis and manuscript preparation. TY performed the data analyses and wrote the manuscript. HX helped perform the analysis with constructive discussions. All authors contributed to the article and approved the submitted version.This study was supported by the National Natural Science Foundation of China (No. 11871077), the NSF of Anhui Province (1808085MA04 and 1908085MC62), the NSF of Anhui Provincial Department of Education (KJ2020A0894 and KJ2021A0650), Graduate Scientific Research Project of Anhui Provincial Department of Education (YJS20210515), Research and Innovation Team of Hefei Preschool Education College (KCTD202001), and Graduate offline course graph theory (2021aqnuxskc02).The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.


R-submodule and α-intuitionistic fuzzy (α-IF) submodule. The proof that the level set of LIF-3 is contained in the level set of α-IF is given, and it is exclusively employed to define linear codes over α-IF submodule. Further, α-IF cyclic codes are presented along with their fundamental properties. Finally, an application based on genetic code is presented, and it is found that the technique of defining codes over α-IF submodule is entirely applicable in this scenario. More specifically, a mapping from the ℤ64 module to a lattice L (comprising 64 codons) is considered, and α-IF codes are defined along with the respective degrees.In the last few decades, the algebraic coding theory found widespread applications in various disciplines due to its rich fascinating mathematical structure. Linear codes, the basic codes in coding theory, are significant in data transmission. In this article, the authors' aim is to enlighten the reader about the role of linear codes in a fuzzy environment. Thus, the reader will be aware of linear codes over lattice valued intuitionistic fuzzy type-3 (LIF-3)  𝔽2, among which the binary codes are the simplest. Say, in a binary field 𝔽2, two codewords 110 and 111 are considered; then, both codewords comprise three bits. Now, when a message carrying 111 is sent in binary coding, the received message should contain 111 theoretically. Often, the transmission is corrupted, and the received message could be carrying 110. This error can be corrected by understanding the coding theory, and data transmission can be made more reliable and accurate. Linear codes are the most basic error-correcting codes which are the subspaces of a vector space. These codes are crucial to data transmission and data storage . The choice of linearization creates a problem; to overcome this drawback, it is replaced by Lattice homomorphism α : L⟶, and it is said to be LIFS-3.Now, when dealing with the data transmission, one cannot always get crisp zeros and ones with data transmission ambiguity being an often occurring scenario. There may be plenty of reasons behind the emergence of uncertainty. These may be lack of knowledge, chance, imprecision, lack of information, and complexity. The imprecision inherent in the process can be dealt with by using fuzzy logic. Fuzzy sets (IF) are opposite to ordinary sets (crisp sets). In crisp set theory, things are categorized by the values 0 and 1, but the full membership and nonmembership are determined in a fuzzy set. This technique copes mathematically with the vagueness of determining boundaries by assigning grades of membership to the elements. Fuzzy set theory as a generalization of the classical set theory was established by Zadeh  in whichtanassov  proposedd Stoeva  proposed⌣ cev i˙  extendedn-dimensional vector space and defined fuzzy codes which are fuzzy subsets of n-tuples over the field. Hamming distance is also defined between two fuzzy codewords. Hall and Dial  is a linearization function satisfying ℓ(f(x))+ℓ(g(x)) ≤ 1.Let S be a non-empty set and L be a complete lattice with top element T and bottom element B. Consider f : S⟶L, g : S⟶L which are membership and nonmembership functions; then, a lattice valued intuitionistic fuzzy set of type-3 (LIFS-3) is the set , where α : L⟶ is a lattice homomorphism with α(T)=1, α(B)=0 satisfyingLet α(f(s))+α(g(s)) ≤ 1 for all s ∈ S.and R is a set equipped with two binary operations, namely, addition + and multiplication · and satisfying the following axioms:R, +) is an Abelian group.⟶ is a semigroup.(a · (b+c)=a · b+a · c and (b+c) · a=b · a+c · a for all a, b, c ∈ R.Multiplication is distributive over addition; that is, Ring theory is one of the extensions of group theory that encompasses a broad set of present study topics in mathematics, computer science, and mathematical/theoretical physics. They have a wide range of applications in the studies of geometric objects and topology, and their connections to other fields of algebra are quite well understood in several contexts. A ring R is called commutative if a · b=b · a for all a, b ∈ R. Consider a commutative ring R. An R-module is a set M together with a binary operation addition ‘⊕' and scalar multiplication ‘∗', where ∗ : R × M⟶M; then, for all r, s ∈ R, a,  b ∈ M, we have the following:r(a ⊕ b)=ra ⊕ rb.r ⊕ s)a=ra ⊕ sa.(rs)∗a=r(s∗a). is a lattice valued intuitionistic fuzzy subset of type-3, where f : M⟶L and g : M⟶L, then A is called an LIF-3 submodule of M if for r ∈ R and a, b ∈ M we have the following:f(a+b) ≥ f(a)∧f(b).f(r.a) ≥ f(a).g(a+b) ≤ g(a)∨g(b).g(r.a) ≤ g(a).If fα, gα : M⟶ be two composition functions, where fα=αof and gα=αog. Then, α-intuitionistic fuzzy (α-IF) submodule if for r ∈ R and x, y ∈ M we have the following:fα(x+y) ≥ fα(x)∧fα(y).fα(r.y) ≥ fα(x).gα(x+y) ≤ gα(x)∨gα(y).gα(r.y) ≤ gα(x).Let IfA=is an LIF-3 submodule, thenα-IF submodule, thenMoreover, if LetA=be an LIF-3 submodule. Then, the necessary and sufficient conditions forAto be anR-module are as follows:f(k1x+k2y) ≥ f(x)∧f(y),g(k1x+k2y) ≤ g(x)∨g(y),x, y ∈ M and k1, k2 ∈ R.∀ An LIF-3 submoduleA=of a ringRis called an LIF-3 ideal if for eachx, y ∈ R,f(x − y) ≥ f(x)∧f(y).f(x.y) ≥ f(x)∨f(y).g(x − y) ≤ g(x)∨g(y).g(x.y) ≥ g(x)∧g(y).LetA=be an LIF-3 submodule; then, the level sets are defined asα -IF submodule In a similar fashion, we can define level sets for an We can also write thatM=ℤpkn which is a ℤpk-module, then an LIF-3 submodule A of M is termed as an LIF-3 linear code having length n over ℤpk.If we consider a module LetA=be an LIF-3 submodule andbe anα-IF submodule. LetAl1l2andbe the level sets forAand, whereα(l1)=t1andα(l2)=t2. Then,Al1l2 and A and m ∈ Al1l2; then, f(m) ≥ l1 and g(m) ≤ l2 imply that fα(m) ≥ α(l1) and gα(m) ≤ α(l2), where α(l1), α(l2) ∈ . From this, we get that m ∈ Aα(l1)α(l2). Thus, Al1l2⊆Aα(l1)α(l2). Let α(l1)=t1 and α(l2)=t2; then, Al1l2⊆At1t2.Let α-IF submodule, so we will use α-IF submodule for further discussion of codes. Let us consider a module M=ℤpkn which is a ℤpk module, and let α-IF submodule; then, α-IF linear code of length n over the module ℤpk.From the above proposition, we have concluded that the level set of LIF-3 submodule is contained in Considerℤ4-module and latticeL={0,1, a, b, c, d, }having 0, 1 as bottom and top elements witha ≤ c, a ≤ d, andb ≤ c. Letf, g : ℤ4⟶Landα : L⟶be defined asα(0)=α(b)=0, α(a)=0.2=α(c), and α(d)=α(1)=1. Then, α -IF ℤ4 module; therefore, α -IF linear code.Let α-IF submodule t ∈ , is said to be the degree of that element t and it is denoted by pt. In example 1, p0=2, p1=2, p0.2=2.Consider an Consider anα-IF submoduleis anα-IF linear code overℤpkiffare linear codes.α-IF submodule and k be non-empty subset of α-IF characteristic function of k is denoted by χk=〈fαχk, gαχk〉, where fαχk, gαχk : M⟶; then,Let Let𝒞be a subset of anα-IF submoduleℤpkn; then,𝒞is anα-IF linear code overℤpkiffχCfor𝒞is anα-IF linear code over the same ringℤpk.𝒞 be a linear code implying that 𝒞 is an α-IF submodule. Now, if x, y ∈ 𝒞, then by χC, we haveLet 𝒞 is an α-IF submodule, then x+y,  r.x ∈ 𝒞 and for all r ∈ R imply that fαχC(x+y)=1=1∧1=fαχC(x)∧fαχC(y) and fαχC(r.x)=1=fαχC(x). Similarly, gαχC(x+y) ≤ gαχC(x)∨gαχC(y) and gαχC(r.x) ≤ gαχC(x). Thus, the χC of 𝒞 is an α-IF submodule; hence, it is an α-IF linear code.As χC be a linear code; hence, it is an α-IF submodule. Let x, y ∈ 𝒞; then, by Conversely, suppose that the fαχC(x+y) ≥ fαχC(x)∧fαχC(y)=1∧1=1 and fαχC(r.x) ≥ fαχC(x)=1. Similarly, gαχC(x+y) ≤ gαχC(x)∨gαχC(y)=0∨0=0 and gαχC(r.x) ≥ gαχC(x)=0; hence, x+y,  r.x ∈ 𝒞; thus, 𝒞 is an α-IF submodule, so 𝒞 is an α-IF linear code.imply that Letbe anα-IFS ofℤpkn. is anα-IF linear code overℤpkiffis anα-IF linear code.Letbe anα-IF fuzzy submodule; then,is also a linear code such that fort1, t2 ∈ , fα(x)=t1andgα(x)=t2; then,Ais said to be a trivialα-IF linear code.LetM=ℤpknbe a module over the ringℤpk. Then, twoα-IF submodulesandare said to be orthogonal ift1, t2 ∈ , we haveM.Furthermore, for all Letbe twoα-IF submodules which are defined asandAsfαt1={a ∈ M : fα(a) ≥ t1}, we can compute the valuesfαt1fort1 ∈ .Similarly, the remaining values given in Hence, LetMbeℤpkmodule andbe twoα-IF submodules. If for allw ∈ M,fα(w)=rorgα(w)=r, wherer ∈ , then setis not orthogonal to setLetbe anα-IF submodule, sois a linear code overℤpkhaving lengthn. Consider setsAs the setℤpkn is finite, im(fα) and im(gα) are also finite. Consider an order t1 ≥ t2 ≥ ⋯≥tr; then, the set im(fα) satisfies this order. Similarly, suppose t1′ ≤ t2′ ≤ …≤tr′ and this order is satisfied by im(gα). Gtjtj′ . Hence, A can be obtained from the matrices Gt1t1′, Gt2t2′,…, Gtrtr′.LetM=ℤpknbe a finite module andbe anα-IF submodule ofM. Then, there is anα-IF submodulesuch thatis orthogonal toif and only if|Im(fα)|, |Im(gα)| > 1, and for anyr ∈ Im(fα), there exists an elementη ∈ Im(fα)withfαr=(fαη)T. Similarly, for any elementp ∈ Im(gα), there existð ∈ Im(gα)such thatgαp=(gαð)T.M be a module and α-IF submodule of M. Suppose that |Im(fα)|=s > 1 and |Im(gα)|=k > 1 and for any element r ∈ Im(fα) there is an element η ∈ Im(fα) such that fαr=(fαη)T. Moreover, for any element p ∈ Im(gα), there exists an element ð ∈ Im(gα) such that gαp=(gαð)T.Now, consider an order for Im(fα) and Im(gα) asLet M asj=1,…, m. Now, if we define an α-IF set fα′, gα′ : M⟶, then for a ∈ M, we have fα′(a)=1 − tm−j+1 and gα′(a)=1 − tm−j+1′.By using these compositions, we can also define sets which form partition of module fα)={t1 ≥ t2 ≥ ⋯≥tm} imply that fαt1⊆fαt2⊆⋯⊆fαtm. Similarly, for Im(gα)={t1′ ≤ t2′ ≤ ⋯∈≤tm′}, we have gαtm⊆⋯⊆gαt2⊆gαtm. As for any element r ∈ Im(fα), there is an element η ∈ Im(α°f) such that fαr=(α°fη)T. Similarly, for any p ∈ Im(gα), there is an element ð ∈ Im(gα) such that gαp=(gαð)T, as in finite module there is a property related to orthogonality by which fαtj=(fαtm−j+1)T. Accordingly, we have fα1−tm−j+1′={a ∈ M : fα′(a) ≥ 1 − tm−j+1}=MjUMj−1U ⋯ UM1=fαtj=(fαtm−j+1)T.As we know, Im(gαtj′=(gαtm−j+1′)T and gα1−tm−j+1′′={a ∈ M : gα′(a) ≥ 1 − tm−j+1′}=M1U … UMj−1UMj=gαtj=(gαtm−j+1′)T. Thus, α-IF submodule.Furthermore, α-IF submodules such that these two sets are orthogonal to each other; then, we have |Im(fα)| > 1 and |Im(gα)| > 1. For all t, t′ ∈ , we also have fα1−t1′=(fαt)T and gα1−t1′′=(gαt′)T; then, for any element r ∈ Im(fα), there exist η ∈ Im(fα) such that fαr=(fαη)T, and for any element p ∈ Im(gα), there exist ð ∈ Im(gα) such that gαp=(gαð)T; this is due to the reason that im(fα)′={1 − t : t ∈ fα} and im(gα′)={1 − t′ : t′ ∈ gα}.Conversely, suppose that Suppose that, andbe threeα-IF fuzzy submodules of a moduleMsuch thatis orthogonal toandis orthogonal to; then,α-IF fuzzy submodules of a module M such that b ∈ fαt1−′, for t ∈ . Then, for a ∈ fαt, 〈a, b〉=0. b ∈ fαt1−″. Thus, fαt1−′⊆fαt1−″. Therefore, fαt″⊆fαt′. By following the same strategy, we get fαt′⊆fαt″. Thus, fα′=fα″. In a similar manner, we can show that gα′ =  gα″. Hence, Let LetMbe a finiteℤpkmodule andbe anα-IF submodule ofM; if there existsα-IFSsuch thatis orthogonal to the submodule, thenis anα-IF submodule of moduleM.Consider twoα-IF linear codesand; then, these twoα-IF codes are said to be equivalent if the level setsand forandare equivalent.α-IF cyclic codes and some important results related to these codes.Cyclic codes have long since been one of the most interesting families of codes because of their rich algebraic structure, and these codes play an important role in coding theory. In this section, we will discuss Consider anα-IF submoduleof moduleℤpkn; then,is called anα-IF cyclic code having lengthnoverℤpkif for each ∈ ℤpknwe haveTheα-IF submoduleis anα-IF cyclic code overℤpkif and only ifare cyclic codes overℤpk.α-IF submodule and i=0,1,…, n − 1 we have Let As which imply that w0, w1,…, wn−1) ∈ ℤpkn such that fα < fα. Suppose t1′=fα. Similarly, t2′=gα imply that Conversely, suppose that Letℤpknbe a module andbe anα-IF submodule.is anα-IF cyclic code on moduleℤpknif and only if the characteristic function of a level setis anα-IF cyclic code onℤpkn.Consider a moduleℤpkn; then,is anα-IF cyclic code on moduleℤpknif and only if for each ∈ ℤpknwe haveℤpkn be a module and α-IF cyclic code on module ℤpkn; then, we haveLet Thus,fα and gα; then, by A is an α-IF cyclic code.Conversely, suppose that the above equality holds for Consider twoα-IF cyclic codesand; then, we have the following:is anα-IF cyclic code.is anα-IF cyclic code.is anα-IF cyclic code.(1)α-IF modules of module ℤpkn such that α-IF cyclic codes corresponding to α-IF modules. Then, for  ∈ ℤpkn,Let In a similar manner, we haveα-IF modules, we get again an α-IF module; thus, α-IF cyclic code.(2)w0, w1,…, wn−1) ∈ ℤpkn,Now, for (gα+gα′) ≤ (gα+gα′). Thus, α-IF cyclic code.Similarly, we can show ((3)Proof follows from (2). By taking intersection of two Ifℤpknis a module, thenis said to be anα-IF cyclic code iff the non-empty level setsareα-IF ideals of the factor ring(ℤpk[X]/(Xn − 1)).ℤpkn and a factor ring (ℤpk[X]/(Xn − 1)). Define a mapping as ϕ : ℤpkn⟶(Zpk[X]/(Xn − 1)) which is also an isomorphism. Let b= ∈ ℤpkn such that ϕ(b)=∑j=0m−1bjXj.Consider a module α-IF cyclic code; this implies that α-IF cyclic code over ℤpk. Cyclic codes are ideals in factor ring, which implies that Suppose that t1, t2 ∈ , the set ϕ, then the level set α-IF cyclic code; thus, α-IF cyclic code over ring ℤpk.Conversely, suppose that for ℤpk is a finite ring and if fα) and im(gα) are also finite; then, we have gjk(X) ∈ ℤpk[X] is the generator polynomial for gj+1k(X)/gjk(X), j=1,…, r − 1.As LetS={g1k(X), g2k(X),…, grk(X)}be a set of polynomials such that the polynomial gj(X) divides Xn − 1 for each j=1,…, r. Ifgj+1k(X)|gjk(X)forj=1,2,…, r − 1andℤpkn=gj+1k(X), then the set of polynomials determines anα-IF cyclic code, wheregjk(X)is the collection of level cut cyclic subcodes ofProof follows from α-IF gray code by using the compositions fα and gα. Consider a map η : ℤ22⟶ℤ22; then, η is called a gray map defined as η(0)=00, η(1)=01, η(2)=11, η(3)=10.The gray code, which is also called reflected binary code, is an ordering of the binary numeral system such that two successive values vary in a single bit. We will define Consider a mappingη : ℤ22⟶ℤ22which is a gray map. Suppose thatS(ℤ22)andS(ℤ22)be twoα-IF fuzzy subsets ofℤ22andℤ22. Forfα, gα ∈ S(ℤ22), anα-IF gray mapη∗ : S(ℤ22)⟶S(ℤ22)is defined asConsider two mappingsfα,  gα : ℤ4⟶defined as follows:By definition, we havea=0,Now, for α-IF gray map corresponding to these mappings are given in The values for ℤpk to ℤppk−1. Using this map, an α-IF generalized gray map can be defined.Now, we consider generalizing fuzzy gray map which is a mapping from A mappingη∗ : S(ℤpk)⟶S(ℤppk−1)is said to be anα-IF generalized gray map if for anyfα,  gα ∈ S(ℤpk)we haveIfis anα-IF code overℤpk, then it is said to be anα-IFℤpk-linear code if it is an image under theα-IF generalized gray map of a linear code overℤpk.Consider anα-IF codeC; then, it is anα-IFℤpkcyclic code ifCis anα-IFℤpk-linear code and also if it is the image under theα-IF generalized gray map of a cyclic code overℤpk.Consider mappingsfα, gα : ℤ26⟶whereandThen,C= is an α -IF linear code having length six over ℤ2 . Now, if we consider another code where fα,  gα : ℤ43⟶ defined asC= is an α -IF linear code having length 3 over ℤ4.Anα-IF gray mapη∗is a bijective map.η∗ is a one-to-one function.This is due to the fact that The coding theory deals with encoding and decoding a message. During transmission of data, errors may occur; they can be detected and corrected by using different procedures such as parity check, syndrome decoding, and redundancy check which are for ordinary codes. Similarly, a method can be adopted in fuzzy codes for the confirmation about the codeword, that is, whether it belongs to the transmitted code or not. This can be done by considering the level set from which one can determine the confirmation degree as to whether a received codeword belongs to the original code. This method can be explained through the following example.Consider a moduleℤ24, whereα-IF submodule where fα, gα : ℤ24⟶ andLet which is a subset ofℤ24and also a linear code. Now, by transmitting this code, the received codeword has errors; assume that the received codewords are{000; .01, 010; 0.001, 011; 0.01, 100; 0.1, 101; 0.01, 110; \\0.1, 001; 0.01, 111; 0.999}.fα, gα corresponding to the codewords of code C.t1 > 0.9 and t2 ≤ 0.01 that the received codeword is in C.The level set is defined as ℤpkm be a module and α-IF submodule. If 𝒞 is an α-IF linear code, then the level set 𝒞t1t2 of 𝒞 is also an α-IF linear code. As the elements in ℤpkm are the words of length m over the alphabet {0,1,2,…, pk − 1} and the level set 𝒞t1t2 also consists of the elements of ℤpkm, then for a member a ∈ 𝒞t1t2 and for any integer r ≥ 0, the sphere of radius r and center a is defined as follows.Suppose Consider a moduleM=ℤpkm; then, the sphere having radiusr(where 0 ≤ r ≤ m))containswords.ℤpkm. Let x be a fixed word in ℤpkm. Then, the number of the words which differ from x in m position is Consider a module LetM=ℤ22, whereℤ22={00,01,10,11}. Here,k=1, p=2, m=2, and0 ≤ r ≤ 2. If we take a wordx=10, then its distances from the remaining codewords ared=1, d=2, d=1, and the sum of these distances is equal to 4. Now,, which is also equal to 4.Amino acid sequence of a protein can be determined from the sequence or order of DNA and RNA molecules . Althougℤ4. Bennenni et al. . Suppose f, g : ℤ64⟶L where L is a lattice consisting of 64 codons; the values for these functions and their composition are shown in α-IF submodule so it is an α-IF linear code. We will use the following methodology.In the presented study, codon structure is investigated by involving the lattice structure of codons along with module over a ring. As there are 64 possible codons, we take L comprises 64 codons and a module Z64={0,1,…, 63} which also consists of 64 elements. Let a=0 ∈ Z64; we know that f, g : Z64⟶L. Then, f(0)=UUU, g(0)=AAA, which are top and bottom elements of the lattice L as shown in , assign the values to these functions as fα(0)=1 ∈ , gα(0)=0 ∈ , where fα(0)+gα(0) ≤ 1. Similarly, values can be assigned to the remaining codons, which is shown in α-IF submodule with module Z64 is referred to as a linear code. By using t ∈ ; for example, for t=1, the degree is 1 and for t=0.3 the degree is 10. The remaining values can be computed in a similar way. Degrees of respective codons are also given in Consider the lattice shown in . As fα, U show hydrophobic amino acids, and those with second base A show hydrophilic amino acids. There are total 16 codons that specify hydrophobic and hydrophilic amino acids, so we can define an α-IF linear code by considering the ℤ16 module. Let f, g : ℤ16⟶L and fα, gα : ℤ16⟶, where lattice L consists of 16 elements. A comparison can be made between the two properties of amino acid by employing the concept of degree given in The most important properties of amino acids are their hydrophobic and hydrophilic properties. As in lattice diagram, codons with second base α-IF submodule are defined. These two structures are related by means of their level sets; that is, the level set of LIF-3 submodule is contained in the level set of α-IF submodule. After obtaining this result, further codes are discussed over α-IF submodule. Linear codes and cyclic codes are defined over α-IF submodule. Some important properties and results related to these codes are investigated. It is also concluded that this concept of α-IF linear codes is entirely applicable in genetic code. There are 64 codons that specify different amino acids, so Z64 module is considered here and α-IF linear codes are defined over Z64 module.Communication systems are designed for data transmission, working on the principle of encoding and decoding information. In classical coding theory, different procedures can be adopted to detect and correct errors that may arise during communication. The communication process is highly influenced by vagueness, inaccuracy, imprecision, and uncertainty. The classical coding methodologies are sometimes not efficient in handling such situations. Therefore, fuzzy logic is a viable option to handle such type of information. Lattice valued intuitionistic fuzzy sets are the generalization of basic fuzzy sets that incorporated the degree of both membership and nonmembership. Thus, they constitute a more efficient framework to model uncertain data. In this article, LIF-3 submodule and α-IF submodule by involving ℤpkn module over a ring ℤpk instead of the field 𝔽. Moreover, there are several generalizations of fuzzy sets like picture fuzzy sets [α-IF codes are defined by considering the modules Z64 and Z16. This methodology seems to be attractive for further investigation of genetic codes.In literature, conventional codes such as hamming codes and Hadamard codes are defined, which are constructed by considering vector spaces and fields. Modules are the extension of vector spaces. Thus, this work can be extended further by employing the present methodology and by using modules instead of vector spaces and fields. This can be done by investigating these codes over zzy sets , Pythagozzy sets , hesitanzzy sets , and neuzzy sets  where th


Electromechanical phonon-cavity systems are man-made micro-structures, in which vibrational energy can be coherently transferred between different degrees of freedom. In such devices, the energy transfer direction and coupling strength can be parametrically controlled, offering great opportunities for both fundamental studies and practical applications such as phonon manipulation and sensing. However, to date the investigation of such systems has largely been limited to linear vibrations, while their responses in the nonlinear regime remain yet to be explored. Here, we demonstrate nonlinear operation of electromechanical phonon-cavity systems, and show that the resonant response differs drastically from that in the linear regime. We further demonstrate that by controlling the parametric pump, one can achieve nonlinearity-mediated digitization and amplification in the frequency domain, which can be exploited to build high-performance MEMS sensing devices based on phonon-cavity systems. Our findings offer intriguing opportunities for creating frequency-shift-based sensors and transducers. Electromechanical phonon cavity systems offer advantages for solid state implementation, but thus far investigation has been limited to the linear regime. Here, Miao et al demonstrate non-linear operation of an electromechanical phonon cavity system and the drastic difference to the linear regime. As the degree of freedom (DOF) in the system increases , nonlinearity gives rise to unique phenomena such as synchronization8, chaos9, internal resonance10, and generation of frequency comb13, by enabling and modulating coherent energy transfer between the different DOFs14.Nonlinearity is ubiquitous in real-world physical systems. Among the many man-made structures, resonant microelectromechanical and nanoelectromechanical systems (MEMS/NEMS) offer great opportunities for designing, tuning, and exploiting nonlinearities and have enabled the exploration of nonlinear processes and nonlinearity-dictated device properties, such as energy dissipation15 offer the unique capability of phonon manipulation by parametrically coupling a mechanical resonance to a phonon cavity17, in many aspects analogous to a photon-cavity22, which has enabled a plethora of exotic physical phenomena such as self-cooling24, induced transparency26, parametric amplification27, and quantum squeezing29. These exquisite functions are realized through a pump signal in the cavity’s sideband, which parametrically controls the dynamical coupling and backaction between the resonant mode and the cavity32.Among multi-DOF systems, electromechanical phonon-cavity systems35, such as sensing.One key feature that differentiates the phonon-cavity systems is that they operate entirely in the electromechanical domain, and thus are much more advantageous for implementation using monolithic solid-state devices. Further, the characteristic frequency of the phonon cavity is typically in the same frequency band as the resonant mode , which greatly simplifies the signal transduction, making such systems promising for frequency-shift-based applications14, research to date has been largely confined to linear operations, i.e., with limited vibration amplitude, which has plagued the exploration of nonlinear processes in these systems. Here we study nonlinear operation in a microelectromechanical phonon-cavity system and show that the response clearly differs from that in the linear regime. More importantly, leveraging the nonlinearity-mediated bi-stability, we demonstrate digitization and amplification modes of signal sensing enhancement in the frequency domain, which can be further tuned by the degree of nonlinearity in the vibration response.However, in such parametrically-coupled systems, despite the nonlinear nature of the coupling between the different DOFsω1 = 2π × 6969.1 Hz) and a flexural mode resonance (at frequency ω2 = 2π × 16649.6 Hz). In this work, we specifically design our device to facilitate the observation of dynamical coupling in the nonlinear regime. This is achieved by the small capacitance gap, large capacitance area, and high-quality factor of the resonator (see “Methods”). By activating different combinations of the 12 electrodes , with clear nonlinearity-mediated bi-stability . We denote the critical frequency where such bi-stability occurs as ωb, which depends on the driving amplitude and the sweep direction  and downward (high to low) frequency sweeps Fig. . We now eps Fig. , the effωp has very strong and distinct effects on the resonant response: when pumping in the vicinity of the red side band  as ωp approaches ωred, producing a dip-like feature in the 2D color plot; when pumping around the blue side band  with a sudden jump at ωp ≈ ωblue, resulting in a spike-like feature in the 2D color plot  in the frequency down-sweeps, regardless the position of ωb. This is clearly visible in the 3D plots  are the displacement and natural frequency of the resonant modes, respectively. Specifically, i = 1 corresponds to the mode of interest  being the harmonic driving force), and i = 2 corresponds to the phonon cavity . The energy dissipation rates are given by γi, and cij  give the intra-modal (i = j) and intermodal (i ≠ j) parametric coupling coefficients, with ωp being the frequency of parametric pump. All these terms are normalized using the effective mass of the oscillator. ε is introduced as a scaling parameter, based on the assumption that all terms multiplied by ε are small compared with the terms on the left side of the equations. This assumption is true for steady-state vibration of high Q resonators, where the energy stored in the resonator  is much greater than dissipation or transfer between modes. This mathematical treatment allows us to apply multiple-scale approximation using the standard multidimensional Newton-Raphson algorithm40 (see SI 2.1 and 2.3–2.4 for details), from which we are able to numerically solve the above equations in the vicinity of the cavity sidebands (ωp ≈ ωred and ωp ≈ ωblue), with the results shown in Fig. To understand the unique behavior of the nonlinear response in our phonon-cavity system, we examine its equations of motion by introducing Duffing nonlinearity into this 2-DOF systemωp near ωred, and the spike-like feature for ωp near ωblue. Such unique behavior can be qualitatively understood, in a simplified picture, by considering the nonlinearity-induced amplitude bi-stability in the resonance mode, together with its parametric-pump-controlled coupling to the phonon-cavity: in downward frequency sweeps, the resonant response initially follows the upper branch among the two stable solutions from the softening Duffing equation, before jumping to the lower branch at ωb. When ωp approaches ωred, the parametric pump causes phonon to be removed from the resonant mode . In contrast, when ωp ≈ ωblue, phonons are created in the resonant mode .The numerical results show very good agreement with the experimental data  and frequency (f) of a resonator, leading to an A-f relationship. Second, in phonon-cavity systems, the amplitude of a given mode is affected by the frequency difference between some other modes (such as the pump and the cavity modes), which gives rise to an f-A effect. By operating a phonon cavity device in a nonlinear regime, one can excite both effects, resulting in an f-A-f transduction, enabling signal amplification entirely within the frequency domain. While quantitative explanations for specific details (such as the asymmetry in ωp for the dip-like and spike-like features) require more in-depth analyses and evaluation of multiple expressions derived from Eq. It is also interesting to analyze the observed phenomena by considering two different effects. First, with the sudden jump between its two stable branches, Duffing bi-stability produces a distinct relationship between the amplitude (ω2) can function as the front-end transducer, which converts the input signal to a shift in ω2 .We now show that such tuning of nonlinearity-mediated bi-stability, via parametric coupling to the phonon cavity, can be exploited to realize distinct functions in the frequency domain, offering unique capabilities for sensing and signal transduction. In resonant electromechanical transducers, typically, a change in an input physical quantity  engenders a frequency shift, which is then measured or processed. In our devices, the phonon-cavity , as the input change in ω2 is equivalent to a shift of the operation point , as clearly demonstrated in both experiment  can be continuously and smoothly controlled by adjusting the pump strength Vp and the degree of nonlinearity (varying Vd), as shown in Fig. In the first scenario, we utilize the horizontal red-blue boundary in Fig. ωp slightly below ωblue), outlined by the small circles in Fig. ω2 from the input engenders a continuous and amplified output (change in ωb), for which we define the gain A as:Vp and the degree of nonlinearity (adjusting Vd). We find that deeper nonlinear vibration or stronger parametric coupling can produce larger amplification , even exceeding one order of magnitude (red data series). Second, there is a trade-off between gain and linear dynamic range: a larger gain is associated with a smaller dynamic range. By carefully adjusting Vd and Vp, one can choose the optimal operating condition.In the other scenario, we operate along the sloped red-blue boundary .Vf is applied (blue lines) on the electrodes to shift the frequency of mode 2 (ω2). In practice, the shift of ω2 can be caused by external factors such as temperature, pressure, acceleration, and angular velocity. Here, in order to demonstrate the feasibility of the proposed functions, we use the negative stiffness due to electrostatic force to simulate the frequency shift of mode 2 due to such external factors. To minimize the effect from environment variables, the MEMS resonator is measured in a vacuum with a constant temperature of 30 °C. The frequency output of mode 1 (ωb) is monitored (purple lines) using PLL in addition to the frequency sweep measurements ).The PLL set-up is shown in Fig. Vf to control ω2, and monitor the frequency output ωb using PLL. The PLL data  used in simulation and experiments, the results may not exactly agree quantitatively, but qualitatively all key features and behaviors are successfully reproduced experimentally.To verify the signal sensing enchancement functions, we operate the phonon cavity system in corresponding parameter spaces. Figure Similarly, the experimental results for the red pump case Fig.  agree we42. Our results show that, by harnessing nonlinearity in phonon-cavity system, intriguing opportunities emerge for high-performance frequency-shift-based sensors and transducers50. Specifically, compared with MEMS sensors and transducers using single-DOF nonlinear resonators51 or multiple coupled resonators53, the nonlinear phonon-cavity system in this work could potentially offer greater responsivity, additional control, smaller device footprint, and simplified system design, thus resulting in improved performance in MEMS-based sensing and signal transduction applications.These unique nonlinearity-mediated frequency-shift functions can be exploited for constructing phonon-cavity-based sensors and transducers. It is worth noting that operating devices in the frequency domain have a number of advantages, as “time and frequency are the most accurately measurable of all physical quantities”The electromechanical structure is designed to facilitate coherent energy transfer between the two resonant modes via parametric pump-controlled intermodal coupling. A three-segment beam is used to couple the two masses together, facilitating intermodal coupling structurally. A large mass area and the small capacitive gap in the device design make it easier for the resonator to exhibit nonlinearity, without requiring excessive amplitude. The design of multiple groups of electrodes under the resonator allows us to efficiently adjust the frequency of the different modes and implement dynamic coupling at the same time. The quality factor of the resonator can be effectively controlled by the vacuum packaging process.2. The device is operated under room temperature. The device structure, including its cross-section, and the fabrication process are detailed in Fig. The die embedding the vacuum-sealed MEMS resonators is electrically packaged in a ceramic leadless chip carrier with a pressure of 0.1 Pa. The structure layer is 40-μm-Si and the electrode layer is 6-μm-Si. The gap between the structure layer and the electrode layer is 2 μm. The lateral size of the resonator is about 3 mm × 3 mm. The insulating layer underneath the electrode layer is 2-μm-SiOA number of instruments are used in the measurement. The bias voltages are generated by a low noise voltage source (ITECH IT6233). The drive and pump signals are provided by a two-channel lock-in amplifier (Zurich Instruments HF2LI). The response motion of the resonator is detected by a capacitance–voltage (C/V) converting scheme based on a charge amplifier and measured by the lock-in amplifier. Detailed measurement setup and wiring diagram are available in SI 1.2.Vd = 0.2, 0.4, 0.6, 0.8, 1.0 mVpk and Vd = 1, 2, 3, 4, 5 mVpk, respectively. The results in (c) and (d) are both measured with a frequency downward sweep.The linear and nonlinear responses of both resonant modes are first characterized in the absence of the parametric pump. The measurement condition for the dataset shown in Fig. The coherent energy transfer in the phonon cavity is characterized by scanning the parametric pump frequency near either of the cavity sidebands. The measurement and plotting details for Fig. ωp ≈ ωred , the device is measured with Vd = 0.05 mVpk, Vp = 5.0 Vpk, and plotted using ω1 = 2π × 6886.0 Hz, ωred = 2π × 9592.0 Hz. In the linear regime when ωp ≈ ωblue , the device is measured with Vd = 0.05 mVpk, Vp = 2.0 Vpk, and plotted using ω1 = 2π × 6958.0 Hz, ωblue = 2π × 23,582.0 Hz.In the linear regime when ωp ≈ ωred , the device is measured with Vd = 1.5 mVpk, Vp = 5.0 Vpk. and plotted using ω1 = 2π × 6886.0 Hz, ωred = 2π × 9592.5 Hz. When ωp ≈ ωblue , the device is measured with Vd = 1.0 mVpk, Vp = 1.5 Vpk. and plotted using ω1 = 2π × 6958.0 Hz, ωblue = 2π × 23,575.0 Hz.In the nonlinear regime upward sweep, when ωp ≈ ωred , the device is measured with Vd = 1.5 mVpk, Vp = 5.0 Vpk. and plotted using ω1 = 2π × 6886.0 Hz, ωred = 2π × 9592.5 Hz. When ωp ≈ ωblue , the device is measured with Vd = 1.0 mVpk, Vp = 1.5 Vpk. and plotted using ω1 = 2π × 6958.0 Hz, ωblue = 2π × 23,575.0 Hz.In the nonlinear regime downward sweep, when ωd = 2π × 0.05 Hz, ∆ωp = 2π × 0.2 Hz. Note that the plot center frequencies are slightly adjusted for each panel for easy comparison across the different measurement conditions, in order to best illustrate the key findings. Similarly, the driving and pumping strength for the data presented in each panel are also chosen to best illustrate the key findings.In all measurements in Fig. The numerical codes are constructed using C language. The numerical settings for producing the data in Fig. ωp ≈ ωred (a), the device response is calculated with Vd = 0.05 mVpk, Vp = 5.0 Vpk, and plotted using ω1 = 2π × 6886.0 Hz, ωred = 2π × 9592.0 Hz. In the linear regime when ωp ≈ ωblue (b), the device response is calculated with Vd = 0.05 mVpk, Vp = 2.0 Vpk, and plotted using ω1 = 2π × 6958.0 Hz, ωblue = 2π × 23,582.0 Hz.In the linear regime when ωp ≈ ωred (c), the device response is calculated with Vd = 1.5 mVpk, Vp = 5.0 Vpk, and plotted using ω1 = 2π × 6886.0 Hz, ωred = 2π × 9592.5 Hz. When ωp ≈ ωblue (d), the device response is calculated with Vd = 1.0 mVpk, Vp = 2.0 Vpk, and plotted using ω1 = 2π × 6958.0 Hz, ωblue = 2π × 23,575.0 Hz.In the nonlinear regime upward sweep, when ωp ≈ ωred (e), the device response is calculated with Vd = 1.5 mVpk, Vp = 5.0 Vpk. and plotted using ω1 = 2π × 6886.0 Hz, ωred = 2π × 9592.5 Hz. When ωp ≈ ωblue (f), the device response is calculated with Vd = 1.0 mVpk, Vp = 2.0 Vpk, and plotted using ω1 = 2π × 6958.0 Hz, ωblue = 2π × 23,575.0 Hz.In the nonlinear regime downward sweep, when ωd = 2π × 0.05 Hz, ∆ωp = 2π × 0.2 Hz. Note that the plot center frequencies are slightly adjusted for each panel for easy comparison across the different measurement conditions, in order to best illustrate the key findings. Similarly, the driving and pumping strength for the data presented in each panel are also chosen to best illustrate the key findings.In all calculations presented in Fig. The numerical settings for producing the data in Fig. Vd = 0.05 mVpk, Vp = 2.0 Vpk, ω1 = 2π × 6958.0 Hz, and ωp = 2π × 23,582.0 Hz. The frequency steps are ∆ωd = 2π × 0.05 Hz, and ∆ω2 = 2π × 0.2 Hz. The line plots  are taken from data in (b) with ∆ω2 = 0 (c) and ∆ω2 = 2π × 1.0 Hz (d).In the linear operation (b), Vd = 1.0 mVpk, Vp = 2.0 Vpk, ω1 = 2π × 6958.0 Hz, and ωp = 2π × 23,575.0 Hz. The frequency steps are ∆ωd = 2π × 0.05 Hz, and ∆ω2 = 2π × 0.2 Hz. The line plots  taken from data in (d) when ∆ω2 = 0 (g) and ∆ω2 = 2π × 1.0 Hz (h).In the nonlinear operation with downward sweep (f), Vd = 3.0 mVpk, Vp = 0.6 Vpk, ω1 = 2π × 6929.52 Hz, and ωp = 2π × 23,552.51 Hz. The frequency steps are ∆ωd = 2π × 0.01 Hz, ∆ω2 = 2π × 0.01 Hz.In the zoom-in plot (i), Vp = 1.0, 1.5, 2.0 Vpk when Vd is scanned from 2.0 mVpk to 3.0 mVpk. The step of Vd is 0.1 mVpk.The data of digitization in (j) are calculated for Vd is 3.0 mVpk and Vp = 0.35, 0.45, 0.50 Vpk. The step of ∆ω2 is 2π × 0.01 Hz.The data of amplification in (k) are calculated for Vp is 0.50 Vpk and Vd = 2.7, 2.9, 3.0 mVpk. The step of ∆ω2 is 2π × 0.01 Hz.The data of amplification in (l) are calculated for The measurement settings for producing Fig. Vd = 1.25 mVpk, Vp = 1.5 Vpk, ω1 = 2π × 6538.0 Hz, and ωp = 2π × 23,048.0 Hz. The steps are ∆ωd = 2π × 0.02 Hz, and ∆Vf = 0.05 Vpk.In the nonlinear operation at blue pump with downward sweep (a), Vd = 1.25 mVpk, Vp = 1.5 Vpk, and ωp = 2π × 23,048.0 Hz. Vf is scanned from 0 to 1 Vpk and the step of Vf is 0.05 Vpk.When using PLL to test the phonon-cavity system at blue pump (b), Vp = 1.5 Vpk when Vd is scanned from 0.75 mVpk to 1.25 mVpk. The step of Vd is 0.05 mVpk.The experiment of digitization at blue pump in (c), Vd = 1.25 mVpk, Vp = 1.5 Vpk, and ωp = 2π × 23,048.0 Hz. Vf is scanned from 0.445 to 0.495 pk and the step of Vf is 5 mVpk.The experiment of amplification at blue pump in (d), ω2 and output signal ωb under different operation conditions: when there is no parametric pump when there is a blue pump and when there is a red pump applied to the phonon-cavity system.The experiment result of frequency noise is shown in Fig. σ is shown in Fig. The time-domain data is shown in Fig. Supplementary InformationPeer Review File


G = (V(G), E(G)) be a graph with no loops, numerous edges, and only one component, which is made up of the vertex set V(G) and the edge set E(G). The distance d between two vertices u, v that belong to the vertex set of H is the shortest path between them. A k-ordered partition of vertices is defined as β = {β1, β2, …, βk}. If all distances d are finite for all vertices v ∈ V, then the k-tuple , d, …, d) represents vertex v in terms of β, and is represented by r(v|β). If every vertex has a different presentation, the k-partition β is a resolving partition. The partition dimension of G, indicated by pd(G), is the minimal k for which there is a resolving k-partition of V(G). The partition dimension of Toeplitz graphs formed by two and three generators is constant, as shown in the following paper. The resolving set allows obtaining a unique representation for computer structures. In particular, they are used in pharmaceutical research for discovering patterns common to a variety of drugs. The above definitions are based on the hypothesis of chemical graph theory and it is a customary depiction of chemical compounds in form of graph structures, where the node and edge represent the atom and bond types, respectively.Let  L = {l1, l2, …, lk} is a graph's ordered set of vertices and v ∈ G, then the k-tuple r(v|L) = , r, …, r). The notation r is the representation of v with regard to L, and the symbol L is said to be a resolving set if the different vertices of G have different representations regard to L. H's metric dimension, indicated by dim(H), is the minimal number of vertices in the resolving set. The task of computing a graph's locating set is a Non-deterministic Polynomial time problem or NP-hard  = {d, d, …, d} are named as k-tuple representations. If each v in V(G) has a unique representation with regard to β, then the resolving partition of the vertex set is termed β, and the least value of the resolving partition set of V(G) is called the partition dimension of G and is indicated as pd(G)  = {1, 2, 3, …, n} has E(H) = {, 1 ≤ x ≤ y ≤ n}, iffy − 1 = tq for some q, 1 ≤ q ≤ p  = 2 if and only if G = Pn”.Theorem 2 Chartrand et al. (ϕ be a resolving partition of ε(⋎) and ϵ1, ϵ2 ∈ ε(⋎). If d = d for all vertices w ∈ ε(⋎)\, then ϵ1, ϵ2 belong to different classes of ϕ.”d et al.  “Let ϕ bpd of Toeplitz graph with two generators 1 and t in Section 2 and Toeplitz graph partition dimension with three generators 1, 2, and t in Section 3.This study's findings the pd of the Toeplitz graph Tn〈1, t〉, for t ≥ 2 the pd of the graph is three.The coming section is containing the discussion on the Theorem 2.1. A Toeplitz graph withn ≥ 4 isTn〈1, 2〉. After that, pd = 3.Proof. Let the Toeplitz graph with n ≥ 4 is Tn〈1, 2〉 Then we will show that the Toeplitz graph with generators 1 and 2 consist a resolving partition set, β = {β1, β2, β3} with three elements, where β1 = {v1}, β2 = {vk}k≡0(mod 2), β3 = {vk}k≡1(mod 2). Let β = {β1, β2, β3} resolve the vertices of graph G with V(G) = β1 ∪ β2 ∪ β3.k = 1, 2, …, n. In terms of resolving partition set β, we have the following representations of vk.When Because all of the representations of different vertices are distinctpd ≥ 3. Suppose on contrary that pd = 2. We know that pd(G) = 2, iff G is a path graph by Theorem 1, it is not possible for Tn〈1, 2〉. Thus,Conversely: Now, we will show that Hence, from Inequalities (2) and (3), we haveTheorem 2.2. Let a Toeplitz graphTn〈1, 3〉 withn ≥ 5. Thenpd = 3.Proof. Let a Toeplitz graph Tn〈1, 3〉 with n ≥ 5. We will show that the Toeplitz graph with generators 〈1, 3〉 consist of a resolving partition set, β = {β1, β2, β3} with three elements, where β1 = {v1}, β2 = {v2, …, vt}, β3 = {vt+1, …, vn}. There are two cases for β:k ≤ 3, then we can write the representation of vk with respect to β asCase 1: If 1 ≤ Tn〈1, 3〉.where k ≤ n, then we can write the representation of vk with respect to β aCase 2: If 4 ≤ k − 1 ≡ j(mod 3), this shows that all the representations are different, thus,where pd ≥ 3. On contrary suppose that pd = 2. Theorem 1 demonstrates that pd(G) = 2, iff G is a path graph, then it is not possible for Tn〈1, 3〉. Thus,Conversely: We will prove that Hence, from Inequalities (4) and (5), we haveTheorem 2.3. Let a Toeplitz graph with notationTn〈1, t〉 with even generatort ≥ 4, n ≥ t+2. Thenpd = 3.Proof. Let a Toeplitz graph with notation Tn〈1, t〉 with even generator t ≥ 4, n ≥ t+2. The Toeplitz graph with generators 〈1, t〉 consisting of a resolving partition set will be demonstrated. β = {β1, β2, β3}, where β1 = {v1}, 3 = {∀vk|vk ∉ β1, β2}. There are three cases with respect to vk, which are the following;vk with regard to resolving partition set β;Case 1: When where vk with respect to resolving partition set β;Case 2: When z = 1 when z = 0.where vk with respect to resolving partition set β;Case 3: When z = 1 when k = 1 and otherwise z = 0.where It is clear that no two vertices have the same representation, implying that there are not any two vertices with the same representation.pd ≥ 3. Suppose on the contrary that pd = 2. We know that by Theorem 1, it is not possible for even t of graph Tn〈1, t〉. Thus,On contrary, we shall now demonstrate that Hence, from Inequalities (6) and (7), we haveTheorem 2.4. Let a Toeplitz graphTn〈1, t〉 with oddt ≥ 5, n ≥ t+2. Thenpd = 3.Proof. Let a Toeplitz graph Tn〈1, t〉 with odd t ≥ 5, n ≥ t+2. We will show that the Toeplitz graph with generators 1 and t, consists of a resolving partition set, β = {β1, β2, β3} with three elements, where β1 = {v1, v2, vt+1}, 3 = {∀vk|vk ∉ {β1, β2}}.1There are two cases for βwhere 2There are two cases for βwhere 3, we have the following valuesFor β1, β2, and β3From all these cases of βWe conclude that all representations are unique, and no two vertices have identical representations.pd ≥ 3. Suppose on the contrary that pd = 2. We know that by Theorem 1, it is not possible for odd t of graph Tn〈1, t〉. SoIn contrary, we shall now demonstrate that Hence, from Inequalities (8) and (9), we can say thatTn〈1, 2, t〉. If t = 3, 4, 5, and t = 2i, i ≥ 3, n ≥ t + 2 then partition dimension is 4.In this section, we are going to discuss the partition dimension of Theorem 3.1. LetTn〈1, 2, t〉 be a Toeplitz graph. Thenpd = 4.Proof. We split our theorem into three cases.t = 3, 4, 5.Case A: When Tn〈1, 2, t〉 be a Toeplitz graph with t = 3, 4, 5, n ≥ t+2, then we will show that vertices of the Toeplitz graph with three generators consist of a resolving partition set, β = {β1, β2, β3, β4} where β1 = {v1}, β2 = {v2}, β3 = {v3, …, vt}, and β4 = {vt+1, …, vn}. Then there are the three cases that follow:Let k≡1(mod t), then we can write the unique position of vk regarding β as;Case 1: If Tn〈1, 2, t〉where k≡2, 3(mod t), we can write the representations of vk regarding β as;Case 2: If Tn〈1, 2, t〉.where k ≡ 4, 5(mod t), we can write the representations of vk with respect to β asCase 3: If where t = 6, 8.Case B: When 1, β2, β3, β4} be a resolving partition set. Where β1 = {v1}, β2 = {v2, vt}, β3 = {v3, …, vt−2}, and β4 = {vt−1, vt+1, …, vn}. We have different cases on vk, which are following;Let β = {β1;There are two cases for βz1 = 1 when k = even, z = 1 when k ≡ 0(mod 8) and otherwise both are 0.where 2;There are three cases for βz = 0 when 5 ≤ k ≤ t−1 and otherwise z = 2, and z1 has defined in β1.where 3;There are three cases for β4;There is the only case for βIt is clear that no two vertices have the same representation, implying that there are no two vertices with the same representation.t = 2i, i ≥ 5.Case C: When 1, β2, β3, β4} be a resolving partition set. Where β1 = {v2}, β2 = {v6}, β3 = {va}, and β4 = {∀vk|vk ∉ β1, β2, β3}. where Let β = {βvk with regard to the resolving partition set β.The following is a representation of all vertices 1;There are two cases for β2;There are four cases for β3;There are four cases for βz1 = 0 when k = even, otherwise 1 and z2 = 1 when k = odd, otherwise 0.where 4;There is the only case for βIt is clear that no two vertices have the same representation, implying that there does not exist two vertices with the same representation.Converse A, B, and C:pd ≥ 4. On contrary, suppose that pd = 3.We will show that pd is 3. If n consists of sets of different resolving partition set that are following:Different cases on behalf of our assumption that 1 = {v1, v2}, β2 = {v3, v4}, Case 1: β1 = {v1, v3}, β2 = {v2, v4}, Case 2: β1 = {v1, v2, v3}, β2 = {v4}, Case 3: β1 = {v1, v2}, β2 = {v3}, Case 4: β1 = {v1}, β2 = {v2, v3}, Case 5: β1 = {v1, v2, v4}, β2 = {v3}, Case 6: β1 = {v1, v3, v4}, β2 = {v2}, Case 7: β1 = {v1}, β2 = {v2, v4}, Case 8: β1 = {v1}, β2 = {v2, v3, v4}, Case 9: β1 = {v1}, β2 = {v2, v3, v5}, Case 10: β1 = {v1, v5}, β2 = {v2, v4}, Case 11: β1 = {v1, v4}, β2 = {v3, v5}, Case 12: β1 = {v1, v4}, β2 = {v2, v5}, Case 13: β1 ≡ 1(mod 3), β2 ≡ 2(mod 3), β3 ≡ 0(mod 3), then we have the following different vertices with same representation; Case 14: βTn〈1, 2, t〉 into three resolving partition sets. Thus,According to the above cases, we can easily conclude that our supposition is wrong, and we can not resolve the vertices of Hence, from Inequalities (1-3) and (13), we can say thatpd=3, where t ≥ 2 and if Toeplitz graph consists of three generators, pd) = 4, where t = 3, 5 and t = 2i and i ≥ 2.In this study, we looked at different families of Toeplitz graphs and established that the partition dimension of each family is the constant, if the Toeplitz graph consists of two generators, then In this paper, inequality (1) also satisfied the metric dimension results  = 1, is constant, bounded or unbounded?The partition dimension of the Toeplitz graph with two generators Open Problem 2.k ≥ 2, s ≥ 3, t ≥ 4 and gcd = 1, is constant, bounded or unbounded?The partition dimension of the Toeplitz graph with three generators Open Problem 3.If the generators of the Toeplitz graph are increasing then the partition dimension either increasing or decreasing ?The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding authors.RL conceived of the presented idea. AK developed the theory and performed the computations. AA and MA verified the analytical methods. GI and MN investigated and supervised the findings of this work. All authors discussed the results and contributed to the final manuscript.This study was supported by the National Science Foundation of China (11961021 and 11561019), Guangxi Natural Science Foundation (2020GXNSFAA159084), and Hechi University Research Fund for Advanced Talents (2019GCC005).The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.


In this DFT‐mechanistic work, the origin of the observed ring‐size effect is examined in detail using 1‐hexene, CH2=CH2 and H2 as substrates. It is shown that, at room temperature, both the N,N’,N’’,N’’’,N’’’’‐stabilized dimer and the monomer are not coordinated by THF in solution, while the corresponding N,N’,N’’,N’’’‐stabilized structures are coordinated by one THF molecule mimicking the fifth N‐coordination. Catalytic 1‐alkene hydrogenation may occur via anti‐Markovnikov addition over the terminal Ca−H bonds of transient monomers, followed by faster Ca−C bond hydrogenolysis. The higher catalytic activity of the larger N,N’,N’’,N’’’,N’’’’‐stabilized dimer is due to not only easier formation of but also due to the higher reactivity of the catalytic monomeric species. In contrast, despite unfavorable THF‐coordination in solution, the smaller N,N’,N’’,N’’’‐stabilized dimer shows a 3.2 kcal mol−1 lower barrier via a dinuclear cooperative Ca−H−Ca bridge for H2 isotope exchange than the large N,N’,N’’,N’’’,N’’’’‐stabilized dimer, mainly due to less steric hindrance. The observed ring‐size effect can be understood mainly by a subtle interplay of solvent, steric and cooperative effects that can be resolved in detail by state‐of‐the‐art quantum chemistry calculations.Recently, it was shown that the double Ca−H−Ca‐bridged calcium hydride cation dimer [LCaH The catalytic activity of double Ca−H−Ca‐bridged calcium hydride cation dimer complexes is influenced by using macrocyclic tetra‐ or pentadentate ligands. Extensive dispersion‐corrected DFT calculations disclose that this influence is due to a subtle balance of solvent, steric and dinuclear cooperative effects that also depend on the substrate. Further hydrogenolysis of the Ca−C bond of +mA is kinetically 4.1 kcal mol−1 more favorable (via +mTS2) than the preceding alkene addition step and is −21.7 kcal mol−1 exergonic to release the hexane product CH3CH2Bu along with +1 m ⋅ THF, followed by exergonic dimerization of two +1 m ⋅ THF complexes to regenerate the catalyst 2+1 ⋅ THF. The catalytic hydrogenation of CH2=CHBu via this monomeric mechanism is −24.4 kcal mol−1 exergonic over a sizeable barrier of 24.6 kcal mol−1, consistent with the moderate heating at 60 °C required experimentally.2=CHCye (Cye=cyclohexenyl) substrate.2=CH2) is used as substrate instead, the overall catalytic hydrogenation becomes −27.6 kcal mol−1 over a 5.4 kcal mol−1 lower barrier of 19.2 kcal mol−1, suggesting a much faster reaction even at room temperature. Due to reduced steric hindrance, direct CH2=CH2 addition to a Ca−H−Ca bridge of cation dimer 2+1 becomes kinetically only 0.5 kcal mol−1 less favorable . For comparison, similar addition of CH2=CHBu and CH2=CH2 to a Ca−H−Ca bridge of dimer cation +3 encounters relatively higher barriers of 26.3 and 21.8 kcal mol−1 , respectively, consistent with the lower catalytic activity of +3 compared to 2+1 ⋅ THF as observed experimentally.As shown in Figure 2+2 stabilized by larger L5 ligands, the anti‐Markovnikov‐selective addition of CH2=CHBu to the terminal Ca−H bond of THF‐free+2 m is still 1.5 kcal mol−1 endergonic over a sizeable barrier of 23.1 kcal mol−1 (via +mTS3) to form the calcium alkyl complex +mB. Further hydrogenolysis of the Ca−C bond of +mB is kinetically 4.9 kcal mol−1 more favorable (via +mTS4) than the preceding alkene addition step and is −19.0 kcal mol−1 exergonic to release the hexane product CH3CH2Bu along with +2 m, followed by exergonic +2 m dimerization to regenerate the catalyst 2+2. Compared with the above 2+1 ⋅ THF‐catalyzed hydrogenation of CH2=CHBu, the 2+2‐catalyzed one is indeed kinetically 1.5 kcal mol−1 more favorable and insensitive to THF coordination, consistent with the higher catalytic activity observed for the 2+2 catalyst stabilized by larger L5 ligands.2=CH2 is used as substrate instead, the overall barrier for catalytic hydrogenation is reduced to only 17.5 kcal mol−1 via the cation monomer +2 m, consistent with the rapid reaction observed even at 0 °C.2=CH2 addition to a Ca−H−Ca bridge of L5‐stabilized 2+2 encounters a high free energy barrier of 36.6 kcal mol−1 and thus is kinetically highly disfavored. Moreover, compared with 2+1 ⋅ THF‐catalyzed hydrogenation of CH2=CHBu, the 1.5 kcal mol−1 lower barrier computed for the catalyst 2+2 is due to not only the 0.9 kcal mol−1 lower free energy for the monomer +2 m formation, but also to the 0.6 kcal mol−1 lower alkene addition barrier to THF‐free+2 m than to THF‐coordinated +1 ⋅ THF. In other words, the observed “ligand ring‐size” effectAs shown in Figure 2 isotope exchange, the dimeric mechanism becomes more important due to evidently reduced steric hindrance and stronger dinuclear cooperative effects. As seen in Figure 2 isotope exchange may occur via a cooperative Ca−H−Ca bridge of THF‐free cation dimer 2+1 over a low barrier of 16.9 kcal mol−1 (via 2+TS5) after THF elimination,2 isotope exchange via a Ca−H−Ca bridge of THF‐coordinated 2+1 ⋅ THF becomes kinetically 5.0 kcal mol−1 less favorable (via 2+TS5a), mainly due increased steric hindrance. On the other hand, the cation monomer +1 m ⋅ THF is 7.8 kcal mol−1 higher in free energy but intrinsically 8.9 kcal mol−1 more reactive than 2+1 ⋅ THF, and thus is kinetically 1.1 kcal mol−1 more favorable for catalytic H2 isotope exchange (via +mTS5). However, 2+1 ⋅ THF remains kinetically 3.9 kcal mol−1 less active than 2+1 for the catalytic H2 isotope exchange. Considering the overall reaction of 2+1 ⋅ THF+H2→2+TS5+THF, it is clear that such H2 isotope exchange is disfavored by the THF coordination by about 2.8 kcal mol−1. For comparison, a similar dimeric mechanism for cation +3‐catalyzed H2 isotope exchange  encounters a moderate but higher free energy barrier of 21.7 kcal mol−1, thus is kinetically less efficient than 2+1.When very small dihydrogen is used as substrate for catalytic H2+2, a similar cooperative Ca−H−Ca‐bridge‐mediated H2 isotope exchange may also occur but over a 3.2 kcal mol−1 higher barrier of 20.1 kcal mol−1 (via 2+TS6). Interestingly, the terminal Ca−H mediated H2 isotope exchange via the THF‐free cation monomer +2 m becomes kinetically competitive over nearly the same barrier of 20.0 kcal mol−1 (via +mTS6). In contrast to the 2+1‐catalyzed H2 isotope exchange that is disfavored by THF coordination within stable 2+1 ⋅ THF, those reactions catalyzed by both the cation dimer 2+2 and the cation monomer +2 m stabilized by the larger L5 ligand are not disfavored by the THF solvent but remain kinetically 3.2 kcal mol−1 less favorable, in sharp contrast to the cases of catalytic hydrogenation of unactivated 1‐alkenes. In other words, no simple “ligand ring‐size” effects can be expected for calcium hydride catalysts, with the catalytic activity actually being due to a subtle balance between solvent, steric and dinuclear cooperative effects.As seen in Figure 2+1 ⋅ THF and 2+2 as catalysts stabilized by macrocyclic L4 and larger L5 ligands and using CH2=CHBu, CH2=CH2 and H2 as substrates of different size, the origin of the experimentally observed ligand ring‐size effect is explored by extensive dispersion‐corrected DFT calculations. It is disclosed that the catalytic activity of calcium hydride cation dimer catalysts can be influenced by a subtle interplay of solvent, steric and substrate‐dependent dinuclear cooperative effects that may change the underlying mechanism in 1‐alkene hydrogenation and H2 isotope exchange reactions.By using cationic dimers solv=3.18 Å). The density‐fitting RI−J approachAll DFT calculations are performed with the TURBOMOLE 7.4 suite of programs.−1 to account for the 1 mol L−1 reference concentration in solution. To check the effects of the chosen DFT functional on the reaction energies and barriers, single‐point calculations at both TPSS‐D3−1 (mean±standard deviation) though as expected 0.5±1.5 kcal mol−1 higher barriers are found at the PW6B95‐D3 level. In our discussion, the more reliable PW6B95‐D3+COSMO‐RS free energies  are used unless specified otherwise. The applied DFT methods in combination with the large AO basis set provide usually accurate electronic energies leading to errors for chemical energies (including barriers) on the order of typically 1–2 kcal mol−1. This has been tested thoroughly for the huge data base GMTKN55More accurate solvation free energies in THF solution are computed with the COSMO‐RS modelThe authors declare no conflict of interest.1As a service to our authors and readers, this journal provides supporting information supplied by the authors. Such materials are peer reviewed and may be re‐organized for online delivery, but are not copy‐edited or typeset. Technical support issues arising from supporting information (other than missing files) should be addressed to the authors.Supporting InformationClick here for additional data file.