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Given a graph X with a Hamilton cycle C, the compression factor κ (X , C) of C is the order of the largest cyclic subgroup of Aut (C) ∩ Aut (X) , and the Hamilton compression κ (X) of X is the maximum of κ (X , C) where C runs over all Hamilton cycles in X. Generalizing the well-known open problem regarding the existence of vertex-transitive graphs without Hamilton paths/cycles, it was asked by Gregor et al. (Ann Comb, arXiv:2205.08126v1, https://doi.org/10.1007/s00026-023-00674-y, 2023) whether for every positive integer k, there exists infinitely many vertex-transitive graphs (Cayley graphs) with Hamilton compression equal to k. Since an infinite family of Cayley graphs with Hamilton compression equal to 1 was given there, the question is completely resolved in this paper in the case of Cayley graphs with a construction of Cayley graphs of semidirect products Z p ⋊ Z k where p is a prime and k ≥ 2 a divisor of p - 1 . Further, infinite families of non-Cayley vertex-transitive graphs with Hamilton compression equal to 1 are given. All of these graphs being metacirculants, some additional results on Hamilton compression of metacirculants of specific orders are also given. [ABSTRACT FROM AUTHOR]

We present a new method for finding k shortest paths between any two vertices in the Cayley graph Cay (G , S) of a finite group G with its generating set S closed under inverses. By using a reduced convergent rewriting system R for G, we first find the lexicographically minimal shortest path between two vertices in Cay (G , S) . Then, by symmetrizing the length-preserving rules of R, we provide a polynomial time algorithm (in the size of certain rewrite rules, the lexicographically minimal shortest path, and k) for finding k shortest paths between two vertices in Cay (G , S) . Our implementation of finding k shortest paths between two vertices in Cay (G , S) is also discussed. [ABSTRACT FROM AUTHOR]

The classical Pauli group can be obtained as the central product of the dihedral group of 8 elements with the cyclic group of order 4. Inspired by this characterization, we introduce the notion of central product of Cayley graphs, which allows to regard the Cayley graph of a central product of groups as a quotient of the Cartesian product of the Cayley graphs of the factor groups. We focus our attention on the Cayley graph C a y (P n , S P n ) of the generalized Pauli group P n on n -qubits; in fact, P n may be decomposed as the central product of finite 2-groups, and a suitable choice of the generating set S P n allows us to recognize the structure of central product of graphs in C a y (P n , S P n ). Using this approach, we are able to recursively construct the adjacency matrix of C a y (P n , S P n ) for each n ≥ 1 , and to explicitly describe its spectrum and the associated eigenvectors. It turns out that C a y (P n , S P n ) is a (3 n + 2) -regular bipartite graph on 4 n + 1 vertices, and it has integral spectrum. This is a highly nontrivial property if one considers that, by choosing as a generating set for P 1 the three classical Pauli matrices, one gets the so-called Möbius-Kantor graph, belonging to the class of generalized Petersen graphs, whose spectrum is not integral. [ABSTRACT FROM AUTHOR]

The spectral decomposition of graph adjacency matrices is an essential ingredient in the design of graph signal processing (GSP) techniques. When the adjacency matrix has multi-dimensional eigenspaces, it is desirable to base GSP constructions on a particular eigenbasis that better reflects the graph’s symmetries. In this paper, we provide an explicit and detailed representation-theoretic account for the spectral decomposition of the adjacency matrix of a weighted Cayley graph. Our method applies to all weighted Cayley graphs, regardless of whether they are quasi-Abelian, and offers detailed descriptions of eigenvalues and eigenvectors derived from the coefficient functions of the representations of the underlying group. Next, we turn our attention to constructing frames on Cayley graphs. Frames are overcomplete spanning sets that ensure stable and potentially redundant systems for signal reconstruction. We use our proposed eigenbases to build frames that are suitable for developing signal processing on Cayley graphs. These are the Frobenius–Schur frames and Cayley frames, for which we provide a characterization and a practical recipe for their construction. [ABSTRACT FROM AUTHOR]

Symmetry properties are of vital importance for graphs. The famous Cayley graph is a good mathematical model as its high symmetry. The normality of the graph can well reflect the symmetry of the graph. In this paper, we characterize the normality of the direct product of Cayley graphs and give a sufficient and necessary condition for the direct product of two graphs to be a Cayley graph. Moreover, a sufficient and necessary condition for judging the normality of the direct product of two graphs is given. [ABSTRACT FROM AUTHOR]

Let Γ = (V (Γ),E(Γ)) be a graph. A subset C of V (Γ) is called a perfect code of Γ, when C is an independent set and every vertex of V (Γ)∖C is adjacent to exactly one vertex in C. Let Γ = Cay(G, S) be a Cayley graph of a finite group G. A subset C of G is called a perfect code of G, when there exists a Cayley graph Γ of G such that C is a perfect code of Γ. Recently, groups whose set of all subgroup perfect codes forms a chain are classified. Also, groups with no proper nontrivial subgroup perfect code are characterized. In this paper, we generalize it and classify groups whose set of all non-perfect code subgroups forms a chain. [ABSTRACT FROM AUTHOR]

Given a graph G , let S and T be two disjoint vertex subsets of equal size k in G. A many-to-many k -disjoint path cover of G joining S and T is a set of k vertex-disjoint paths between S and T that altogether cover every vertex of the graph. It is said to be paired if each vertex of S must be joined to a designated vertex of T , or unpaired otherwise. Let Γ n be a Cayley graph generated by a transposition tree with bipartition V 0 and V 1. Let S ⊆ V 0 and T ⊆ V 1 be two vertex subsets of equal size two. It is shown in this paper that Γ n has an unpaired many-to-many two-disjoint path cover joining S and T with at most n − 4 faulty edges, where n ≥ 4. The result is optimal with respect to the degree of Γ n. As applications, the edge fault-tolerant two-disjoint path covers for the star network, bubble-sort network and modified bubble-sort network are derived. [ABSTRACT FROM AUTHOR]

An (r , z , k) -mixed graph G has every vertex with undirected degree r , directed in- and out-degree z , and diameter k. In this paper, we study the case r = z = 1 , proposing some new constructions of (1 , 1 , k) -mixed graphs with a large number of vertices N. Our study is based on computer techniques for small values of k and the use of graphs on alphabets for general k. In the former case, the constructions are either Cayley or lift graphs. In the latter case, some infinite families of (1 , 1 , k) -mixed graphs are proposed with diameter of the order of 2 log 2 N. [ABSTRACT FROM AUTHOR]

Let s be a positive integer. A graph is s-transitive if its automorphism group is transitive on s-arcs but not on (s + 1)-arcs. In this paper, we study all tetravalent s-transitive graphs of order 6p2. [ABSTRACT FROM AUTHOR]

The sandpile group of a connected graph G, defined to be the torsion part of the cokernel of the graph Laplacian, is a subtle graph invariant with combinatorial, algebraic, and geometric descriptions. Extending and improving previous works on the sandpile group of hypercubes, we study the sandpile groups of the Cayley graphs of F 2 r , focusing on their poorly understood Sylow-2 component. We find the number of Sylow-2 cyclic factors for "generic" Cayley graphs and deduce a bound for the non-generic ones. Moreover, we provide a sharp upper bound for their largest Sylow-2 cyclic factors. In the case of hypercubes, we give exact formulae for the largest n–1 Sylow-2 cyclic factors. Some key ingredients of our work include the natural ring structure on these sandpile groups from representation theory, and calculation of the 2-adic valuations of binomial sums via the combinatorics of carries. [ABSTRACT FROM AUTHOR]

The sandpile group of a connected graph G, defined to be the torsion part of the cokernel of the graph Laplacian, is a subtle graph invariant with combinatorial, algebraic, and geometric descriptions. Extending and improving previous works on the sandpile group of hypercubes, we study the sandpile groups of the Cayley graphs of F 2 r , focusing on their poorly understood Sylow-2 component. We find the number of Sylow-2 cyclic factors for "generic" Cayley graphs and deduce a bound for the non-generic ones. Moreover, we provide a sharp upper bound for their largest Sylow-2 cyclic factors. In the case of hypercubes, we give exact formulae for the largest n–1 Sylow-2 cyclic factors. Some key ingredients of our work include the natural ring structure on these sandpile groups from representation theory, and calculation of the 2-adic valuations of binomial sums via the combinatorics of carries. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cayley regression problem of the cyclic groups and the dihedral groups. We give the sufficient and necessary conditions for the cyclic group ℤn to be a Cayley regression. Moreover, the dihedral group D2n is a 3-Cayley regression if n is a prime power. [ABSTRACT FROM AUTHOR]

For n ≥ 2t + 1 where t ≥ 1, the circulant graph Cn(1, 2, ..., t) consists of the vertices v0, v1, v2, ..., vn-1 and the edges vivi+1, vivi+2, ..., vivi+t, where i = 0, 1, 2, ..., n-1, and the subscripts are taken modulo n. We prove that the metric dimension dim(Cn(1, 2, ..., t)) ≥ ... + 1 for t ≥ 5, where the equality holds if and only if t = 5 and n = 13. Thus dim(Cn(1, 2, ..., t)) ≥ ... + 2 for t ≥ 6. This bound is sharp for every t ≥ 6. [ABSTRACT FROM AUTHOR]

Based on irreducible representations of generalized quaternion groups, closed‐form formulae of Kirchhoff indices and resistance distances between vertex pairs of Cayley graphs on these groups are given. Closed‐form formulae of Kirchhoff indices and resistance distances between vertex pairs of Cayley graphs on quaternion groups are given by using its irreducible representations. [ABSTRACT FROM AUTHOR]

We develop a theory of covering digraphs, similar to the theory of covering spaces. By applying this theory to Cayley digraphs, we build a "bridge" between GLMY-theory and group homology theory, which helps to reduce path homology calculations to group homology computations. We show some cases where this approach allows us to fully express path homology in terms of group homology. To illustrate this method, we provide a path homology computation for the Cayley digraph of the additive group of rational numbers with a generating set consisting of inverses to factorials. The main tool in our work is a filtered simplicial set associated with a digraph, which we call the filtered nerve of a digraph. [ABSTRACT FROM AUTHOR]

V. Levenshtein first proposed the sequence reconstruction problem in 2001. This problem studies the same sequence from some set is transmitted over multiple channels, and the decoder receives the different outputs. Assume that the transmitted sequence is at distance d from some code and there are at most r errors in every channel. Then the sequence reconstruction problem is to find the minimum number of channels required to recover exactly the transmitted sequence that has to be greater than the maximum intersection between two metric balls of radius r, where the distance between their centers is at least d. In this paper, we study the sequence reconstruction problem of permutations under the Hamming distance. In this model we define a Cayley graph over the symmetric group, study its properties and find the exact value of the largest intersection of its two metric balls for d = 2 r . Moreover, we give a lower bound on the largest intersection of two metric balls for d = 2 r - 1 . [ABSTRACT FROM AUTHOR]

It is proved that Tutte's 3-flow conjecture is true for Cayley graphs on groups of order 8p where p is an odd prime. [ABSTRACT FROM AUTHOR]

It is proved that Tutte's 3-flow conjecture is true for Cayley graphs on groups of order 8p where p is an odd prime. [ABSTRACT FROM AUTHOR]

R. Lyons, R. Pemantle and Y. Peres (Ann. Probab. 24 (4), 1996, 1993–2006) conjectured that for a Cayley graph G with a growth rate gr(G) > 1, the speed of a biased random walk exists and is positive for the biased parameter λ ∈ (1, gr(G)). And Gabor Pete (Probability and geometry ´ on groups, Chaper 9, 2024) sheds light on the intricate relationship between the spectral radius of the graph and the speed of the biased random walk. Here, we focus on an example of a Cayley graph, a free product of complete graphs. In this paper, we establish the continuity of the spectral radius of biased random walks with respect to the bias parameter in this class of Cayley graphs. Our method relies on the Kesten-Cheeger-Dodziuk-Mohar theorem and the analysis of generating functions. [ABSTRACT FROM AUTHOR]

In this paper the zero forcing number as well as propagation time of Cay(G, Ω), where G is a finite group and Ω ⊂ G\{1} is an inverse closed generator set of G is studied. In particular, it is shown that the propagation time of Cay(G, Ω) is at most two for some special generators. [ABSTRACT FROM AUTHOR]

The neighbor connectivity refers to the minimum number of vertices whose removal, along with their neighbors, causes a previously connected graph to become disconnected. In this paper we focus on Cayley graphs constructed from the symmetric group Sn. We investigate the bounds of the neighbor connectivity for two cases: when the generating graph is a tree, and when it is a unicyclic graph with a unique cycle of length m, specifically considering cases where m = 3, m = n - 1, or m = n. [ABSTRACT FROM AUTHOR]

Let \(G\) be a quasi-transitive, locally finite, connected graph rooted at a vertex \(o\), and let \(c_n(o)\) be the number of self-avoiding walks of length \(n\) on \(G\) starting at \(o\). We show that if \(G\) has only thin ends, then the generating function \(F_{\mathrm{SAW},o}(z)=\sum_{n \geq 1} c_n(o) z^n\) is an algebraic function. In particular, the connective constant of such a graph is an algebraic number.If \(G\) is deterministically edge-labelled, that is, every (directed) edge carries a label such that no two edges starting at the same vertex have the same label, then the set of all words which can be read along the edges of self-avoiding walks starting at \(o\) forms a language denoted by \(L_{\mathrm{SAW},o}\). Assume that the group of label-preserving graph automorphisms acts quasi-transitively. We show that \(L_{\mathrm{SAW},o}\) is a \(k\)-multiple context-free language if and only if the size of all ends of \(G\) is at most \(2k\). Applied to Cayley graphs of finitely generated groups this says that \(L_{\mathrm{SAW},o}\) is multiple context-free if and only if the group is virtually free.Mathematics Subject Classifications: 20F10, 68Q45, 05C25Keywords: Self avoiding walk, formal language, multiple context free language, Cayley graph, virtually free group

Cayley graph is a graph that encodes the abstract structure of a group. It gives a way of encoding information about a group in a graph. On the other hand, hypergroup is a generalization of group in which the composition of any two elements is a non-empty set. The purpose of this paper is to find a suitable generalization of Cayley graphs to cover hypergroups. More precisely, we introduce generalized Cayley graphs over hypergroups, study their properties and find a simple tool to construct large connected GCH-graphs from smaller GCH-graphs by using graph product. [ABSTRACT FROM AUTHOR]

We show that the category of X-generated E-unitary inverse monoids with greatest group image G is equivalent to the category of G-invariant, finitary closure operators on the set of connected subgraphs of the Cayley graph of G. Analogously, we study F-inverse monoids in the extended signature (· , 1 , - 1 , m) , and show that the category of X-generated F-inverse monoids with greatest group image G is equivalent to the category of G-invariant, finitary closure operators on the set of all subgraphs of the Cayley graph of G. As an application, we show that presentations of F-inverse monoids in the extended signature can be studied by tools analogous to Stephen's procedure in inverse monoids, in particular, we introduce the notions of F-Schützenberger graphs and P-expansions. [ABSTRACT FROM AUTHOR]

We say that a Hamilton cycle C = (x 1 , ... , x n) in a graph G is k-symmetric, if the mapping x i ↦ x i + n / k for all i = 1 , ... , n , where indices are considered modulo n, is an automorphism of G. In other words, if we lay out the vertices x 1 , ... , x n equidistantly on a circle and draw the edges of G as straight lines, then the drawing of G has k-fold rotational symmetry, i.e., all information about the graph is compressed into a 360 ∘ / k wedge of the drawing. The maximum k for which there exists a k-symmetric Hamilton cycle in G is referred to as the Hamilton compression of G. We investigate the Hamilton compression of four different families of vertex-transitive graphs, namely hypercubes, Johnson graphs, permutahedra and Cayley graphs of abelian groups. In several cases, we determine their Hamilton compression exactly, and in other cases, we provide close lower and upper bounds. The constructed cycles have a much higher compression than several classical Gray codes known from the literature. Our constructions also yield Gray codes for bitstrings, combinations and permutations that have few tracks and/or that are balanced. [ABSTRACT FROM AUTHOR]

We study the geometry of compact geodesic spaces with trivial first Betti number admitting large finite groups of isometries. We show that if a finite group G acts by isometries on a compact geodesic space X whose first Betti number vanishes, then diam (X) / diam (X / G) ≤ 4 | G | . For a group G and a finite symmetric generating set S, P k (Γ (G , S)) denotes the 2-dimensional CW-complex whose 1-skeleton is the Cayley graph Γ of G with respect to S and whose 2-cells are m-gons for 0 ≤ m ≤ k , defined by the simple graph loops of length m in Γ , up to cyclic permutations. Let G be a finite abelian group with | G | ≥ 3 and S a symmetric set of generators for which P k (Γ (G , S)) has trivial first Betti number. We show that the first nontrivial eigenvalue - λ 1 of the Laplacian on the Cayley graph satisfies λ 1 ≥ 2 - 2 cos (2 π / k) . We also give an explicit upper bound on the diameter of the Cayley graph of G with respect to S of the form O (k 2 | S | log | G |) . Related explicit bounds for the Cheeger constant and Kazhdan constant of the pair (G, S) are also obtained. [ABSTRACT FROM AUTHOR]

Cayley and nilpotent graphs arise from the interaction between graph theory and algebra and are used to visualize the structures of some algebraic objects as groups and commutative rings. On the other hand, Green and Schroll introduced Brauer graph algebras and Brauer configuration algebras to investigate the algebras of tame and wild representation types. An appropriated system of multisets (called a Brauer configuration) induces these algebras via a suitable bounded quiver (or bounded directed graph), and the combinatorial properties of such multisets describe corresponding indecomposable projective modules, the dimensions of the algebras and their centers. Undirected graphs are examples of Brauer configuration messages, and the description of the related data for their induced Brauer configuration algebras is said to be the Brauer analysis of the graph. This paper gives closed formulas for the dimensions of Brauer configuration algebras (and their centers) induced by Cayley and nilpotent graphs defined by some finite groups and finite commutative rings. These procedures allow us to give examples of Hamiltonian digraph constructions based on Cayley graphs. As an application, some quantum entangled states (e.g., Greenberger–Horne–Zeilinger and Dicke states) are described and analyzed as suitable Brauer messages. [ABSTRACT FROM AUTHOR]

A mixed dihedral group is a group H with two disjoint subgroups X and Y, each elementary abelian of order 2 n , such that H is generated by X ∪ Y , and H / H ′ ≅ X × Y . In this paper, we give a sufficient condition such that the automorphism group of the Cayley graph Cay (H , (X ∪ Y) \ { 1 }) is equal to H ⋊ A (H , X , Y) , where A(H, X, Y) is the setwise stabiliser in Aut (H) of X ∪ Y . We use this criterion to resolve a question of Li et al. (J Aust Math Soc 86:111-122, 2009), by constructing a 2-arc-transitive normal cover of order 2 53 of the complete bipartite graph K 16 , 16 and prove that it is not a Cayley graph. [ABSTRACT FROM AUTHOR]

Let Γ = (V , E) be a graph and a, b nonnegative integers. An (a, b)-regular set in Γ is a nonempty proper subset D of V such that every vertex in D has exactly a neighbours in D and every vertex in V \ D has exactly b neighbours in D. A (0, 1)-regular set is called a perfect code, an efficient dominating set, or an independent perfect dominating set. A subset D of a group G is called an (a, b)-regular set of G if it is an (a, b)-regular set in some Cayley graph of G, and an (a, b)-regular set in a Cayley graph of G is called a subgroup (a, b)-regular set if it is also a subgroup of G. In this paper, we study (a, b)-regular sets in Cayley graphs with a focus on (0, k)-regular sets, where k ≥ 1 is an integer. Among other things, we determine when a non-trivial proper normal subgroup of a group is a (0, k)-regular set of the group. We also determine all subgroup (0, k)-regular sets of dihedral groups and generalized quaternion groups. We obtain necessary and sufficient conditions for a hypercube or the Cartesian product of n copies of the cycle of length p to admit (0, k)-regular sets, where p is an odd prime. Our results generalize several known results from perfect codes to (0, k)-regular sets. [ABSTRACT FROM AUTHOR]

In this paper, we provide a brief introduction to the main notions of geometric group theory and of asymptotic topology of finitely generated groups. We will start by presenting the basis of discrete groups and of the topology at infinity, then we will state some of the main theorems in these fields. Our aim is to give a sample of how the presence of a group action may affect the geometry of the underlying space and how in many cases topological methods may help the determine solutions of algebraic problems which may appear unrelated. [ABSTRACT FROM AUTHOR]

A subset R of the vertex set of a graph $\Gamma $ is said to be $(\kappa ,\tau)$ -regular if R induces a $\kappa $ -regular subgraph and every vertex outside R is adjacent to exactly $\tau $ vertices in R. In particular, if R is a $(\kappa ,\tau)$ -regular set of some Cayley graph on a finite group G , then R is called a $(\kappa ,\tau)$ -regular set of G. Let H be a nontrivial normal subgroup of G , and $\kappa $ and $\tau $ a pair of integers satisfying $0\leq \kappa \leq |H|-1$ , $1\leq \tau \leq |H|$ and $\gcd (2,|H|-1)\mid \kappa $. It is proved that (i) if $\tau $ is even, then H is a $(\kappa ,\tau)$ -regular set of G ; (ii) if $\tau $ is odd, then H is a $(\kappa ,\tau)$ -regular set of G if and only if it is a $(0,1)$ -regular set of G. [ABSTRACT FROM AUTHOR]

Suppose that G is a finite group and S is a non-empty subset of G such that e ∉ S and S -1 ⊆ S. Suppose that Cay(G, S) is the Cayley graph whose vertices are all elements of G and two vertices x and y are adjacent if and only if xy -1 ∈ S. In this paper, we introduce the generalized Cayley graph denoted by Caym(G,S) that is a graph with vertex set consists of all column matrices Xm which all components are in G and two vertices Xm and Ym are adjacent if and only if Xm[(Ym) -1 ] t ∈ M(S), where Ym -1 is a column matrix that each entry is the inverse of similar entry of Ym and M(S) is m × m matrix with all entries in S, [Y -1 ] t is the transpose of Y-1 and m ≥ 1. We aim to determine the structure of Caym(G,S) when G is the dihedral group of order 2n and L S |= 3 for every m ≥ 2, n ≥ 3. [ABSTRACT FROM AUTHOR]

In this paper, we extend our research on Cayley graphs of gyrogroups by exploring connections between the algebraic characteristics of gyrogroups and the combinatorial characteristics of their Cayley graphs. We look into some automorphisms of right Cayley graphs, give sufficient and necessary criteria for a right Cayley graph to be vertex-transitive by left gyrotranslations, and prove a similar result for quotient gyrogroups. Some sufficient and necessary criteria for a left gyroaddition to preserve edge color and for a gyration to be an automorphism on the graph are also given. Lastly, we show that certain Cayley graphs of a gyrogroup encode the gyroaddition table and provide an algorithm to compute the gyroaddition with the aid of the gyration table. [ABSTRACT FROM AUTHOR]

A graph is a core or unretractive if all its endomorphisms are automorphisms. Well-known examples of cores include the Petersen graph and the graph of the dodecahedron—both generalized Petersen graphs. We characterize the generalized Petersen graphs that are cores. A simple characterization of endomorphism-transitive generalized Petersen graphs follows. This extends the characterization of vertex-transitive generalized Petersen graphs due to Frucht, Graver, and Watkins and solves a problem of Fan and Xie. Moreover, we study generalized Petersen graphs that are (underlying graphs of) Cayley graphs of monoids. We show that this is the case for the Petersen graph, answering a recent mathoverflow question, for the Desargues graphs, and for the Dodecahedron—answering a question of Knauer and Knauer. Moreover, we characterize the infinite family of generalized Petersen graphs that are Cayley graphs of a monoid with generating connection set of size two. This extends Nedela and Škoviera's characterization of generalized Petersen graphs that are group Cayley graphs and complements results of Hao, Gao, and Luo. [ABSTRACT FROM AUTHOR]

The exact formula for the average hitting time (HT, as an abbreviation) of simple random walks on Cay (Z N , { ± 1 , ± 2 }) was given by Y. Doi et al.. Y. Doi et al. give a simple formula for the HT's of simple random walks on Cay (Z N , { ± 1 , ± 2 }) by using an elementary method. In this paper, using an elementary method also used by Y. Doi et al., we give a simple formula for HT's of simple random walks on Cay (Z N , { + 1 , + 2 }). [ABSTRACT FROM AUTHOR]

Let ᴦ be a k-regular graph with the second maximum eigen-value λ. Then ᴦ is a Ramanujan graph if λ ≤ 2√k-1. Let G be a finite group and ᴦ = Cay(G; S) be a Cayley graph related to G. The aim of this paper is to investigate the Ramanujan Cayley graphs of sporadic groups. [ABSTRACT FROM AUTHOR]

Perfect state transfer has attracted much interest recently due to its significant applications in quantum information processing and cryptography. In this paper, we investigate perfect state transfer on Cayley graphs over dicyclic groups. Utilizing the representations and characters of dicyclic groups $ T_{4n} $ T 4 n , some necessary and sufficient conditions for Cayley graphs over dicyclic groups admitting perfect state transfer are provided. By those results, some concrete constructions are presented. [ABSTRACT FROM AUTHOR]

Let L be a lattice. For a natural number n, we associate a simple graph, denoted by ΓnL, with Ln\ {0} as the vertex set and two distinct vertices X and Y being adjacent if and only if there exists an n×n lower triangular matrix A over L such that AXT = YT or AYT = XT, where for a matrix B, BT is the matrix transpose of B. In this paper, we study some basic properties of ΓnL. Also, we investigate the genus number of the generalized Cayley graph ΓnL. [ABSTRACT FROM AUTHOR]

Let S be a subset of a finite group Γ. The Cayley signed graph, denoted by Σ = CayΣ(S, σ) has vertex set Γ and two distinct vertices x, y ∈ Γ are joined by an edge from x to y if and only if there exists s ∈ S such that x = sy, where Σ is a signed graph whose underlying graph is Γ and σ : E(Γ) → {+, -} is a function defined as ... In this manuscript, we have characterized the Cayley set and generating sets for which Cayley signed graphs are canonically consistent. [ABSTRACT FROM AUTHOR]

Let R be a finite commutative ring with non-zero identity and Z (R) ∗ be the set of all non-zero zero-divisors of R. In this paper, for a non-empty subset S of Z (R) ∗ , we introduce a new graph, denoted by Γ (R , S) , with vertex set Z (R) ∗ and two distinct vertices x and y are adjacent if and only if there is s ∈ S such that s x = 0 = s y. Then, we investigate some properties of the graph Γ (R , S). [ABSTRACT FROM AUTHOR]

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In this paper, we explore connected Cayley graphs on non-abelian groups of order 5p², where p is a prime greater than 5, and Sylow p-subgroup is cyclic with respect to tetravalent sets that encompass elements with different orders. We prove that these graphs are normal; however, they are not normal edge-transitive, arc-transitive, nor half-transitive. Additionally, we establish that the group is a 5-CI-group. [ABSTRACT FROM AUTHOR]

In this paper, we study the prime order Cayley graph assigned to the group ℤn for different values of n. We specify some of the graph theoretical properties such as chromatic and perfect matching numbers. Furthermore, we determine the adjacency matrices and eigenvalues of the prime order Cayley graph associated with groups ℤn and 𝒟2n.

This paper studies a sliding block puzzle defined on the two by three grid, which we call the 5-puzzle. Our main aim in this paper is to show a fine geometric property of the graph representing the transitions of the positions of the 5-puzzle. We show that by identifying every position with its 180° rotation and applying path contractions, the soccer ball graph is obtained. [ABSTRACT FROM AUTHOR]

Let L (G) denote the space of integer-valued length functions on a countable group G endowed with the topology of pointwise convergence. Assuming that G does not satisfy any non-trivial mixed identity, we prove that a generic (in the Baire category sense) length function on G is a word length and the associated Cayley graph is isomorphic to a certain universal graph U independent of G. On the other hand, we show that every comeager subset of L (G) contains 2 ℵ 0 "asymptotically incomparable" length functions. A combination of these results yields 2 ℵ 0 pairwise non-equivalent regular representations G → Aut (U). We also prove that generic length functions are virtually indistinguishable from the model-theoretic point of view. Topological transitivity of the action of G on L (G) by conjugation plays a crucial role in the proof of the latter result. [ABSTRACT FROM AUTHOR]

Let be a finite group and S be a non-empty subset of A Cayley graph of the group denoted by Cay(S) is defined as a simple graph that its vertices are the elements of and two vertices u and v are adjacent if uv1 2. The minimum edge dominating energy of Cayley graph Cay(; S) is equal to the sum of the absolute values of eigenvalues of the minimum edge dominating matrix of graph Cay(; S). In this paper, we estimate the minimum edge dominating energy of the Cayley graphs for the finite group Sn. [ABSTRACT FROM AUTHOR]

A graph is edge-transitive if its automorphism group acts transitively on the set of edges of the graph. In this paper, we classify hexavalent edge-transitive graphs of order 3 p 2 for each prime p. [ABSTRACT FROM AUTHOR]