Cayley Graphs Research Articles

Vulnerability assessment of a new class of Cayley graph

Vulnerability assessment of a new class of Cayley graph

On large regular [formula omitted]-mixed graphs

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 2log2N.

On the k-Spanning Cyclability of 4-Valent Cayley Graphs on Abelian Groups

A graph X is k -spanning cyclable if for any subset S of k distinct vertices there is a 2-factor of X consisting of k cycles such that each vertex in S belongs to a distinct cycle. In this paper, we examine the k -spanning cyclability of 4-valent Cayley graphs on Abelian groups.

On a family of quasi-strongly regular Cayley graphs

In this paper, we construct a family of quasi-strongly regular Cayley graphs which is defined on a finite group G with respect to a proper subgroup H of G and denoted by Γ H ( G ) . We also compute the full automorphism group and characterize various transitivity properties of Γ H ( G ) for any group G and for any subgroup H of G.

Minimal directed strongly regular Cayley graphs over generalized dicyclic groups

Minimal directed strongly regular Cayley graphs over generalized dicyclic groups

Groups with finitely many Busemann points

We show that an infinite finitely generated group G is virtually \mathbb{Z} if and only if every Cayley graph of G contains only finitely many Busemann points in its horofunction boundary. This complements a previous result of the second named author and M. Tointon.

Binomial Cayley graphs and applications to dynamics on finite spaces

Binomial Cayley graphs and applications to dynamics on finite spaces

The The Cayley Graph of Semi-Direct Product of finite Groups: Interrelationships and Construction

In this paper, we study Cayley graph of the semi-direct product of two finite groups where is an odd prime numbe. Specifically, we endeavor to establish a comprehensive understanding of the Cayley graph by investigating the interrelationships among the constituent elements of the group. Leveraging this acquired knowledge, we systematically construct the Cayley graph, there by contributing to the scholarly discourse on this subject matter.

On the distribution of eigenvalues in families of Cayley graphs

On the distribution of eigenvalues in families of Cayley graphs

Resistance Distance and Kirchhoff Index of Cayley Graphs on Generalized Quaternion Groups

Resistance Distance and Kirchhoff Index of Cayley Graphs on Generalized Quaternion Groups

On invariant generating sets for the cycle space

Consider a unimodular random graph, or just a finitely generated Cayley graph. When does its cycle space have an invariant random generating set of cycles such that every edge is contained in finitely many of the cycles? Generating the free Loop O ( 1 ) O(1) model as a factor of iid is closely connected to having such a generating set for FK-Ising cluster. We show that geodesic cycles do not always form such a generating set, by showing for a parameter regime of the FK-Ising model on the lamplighter group the origin is contained in infinitely many geodesic cycles. This answers a question by Angel, Ray and Spinka. Then we take a look at how the property of having an invariant locally finite generating set for the cycle space is preserved by Bernoulli percolation, and apply it to the problem of factor of iid presentations of the free Loop O ( 1 ) O(1) model.

Fractional revival on Cayley graphs over abelian groups

In this paper, we investigate the existence of fractional revival on Cayley graphs over finite abelian groups. We give a necessary and sufficient condition for Cayley graphs over finite abelian groups to have fractional revival. As applications, the existence of fractional revival on circulant graphs and cubelike graphs are characterized.

Generalized Cayley graphs of complete groups

Generalized Cayley graphs of complete groups

The number of Hamiltonian cycles in groups of Symmetry

Groups are algebraic structures in Abstract Algebra comprised of a set of elements with a binary operation that satisfies closure, associativity, identity and invertibility. Cayley graphs serve as a visualization tool for groups, as they are capable of illustrating certain structures and properties of groups geometrically. In particular, each element in a group is assigned to a vertex in Cayley graphs. By the group action of left-multiplication, distinct elements in the generating sets can act on each element to create varied directed edges (Meier[1]). By contrast, the presence of a Hamiltonian cycles within a graph demonstrates its level of connectivity. In this research paper, utilizing directed Cayley graphs, we present a series of conjectures and theorems regarding the number and existence of Hamiltonian cycles within Dihedral groups, Symmetric groups of Platonic solids and Symmetric groups. By exploring the relationship between Abstract groups and Hamiltonian graphs, this work contributes to the broader field of research pertaining to Groups of Symmetry and Geometric Group Theory.

Hurwitz enumeration problem through the perspective of Cayley graph

The Hurwitz enumeration problem studies how to determine the Hurwitz number for a branch profile, which counts the number of ways a per mutation can be factored into transpositions. In this paper, we consider the length of strictly monotone factorization from the perspective of Cayley graph theory. We represent the factorization problem using a Cayley graph, where vertices are permutations and edges are transpositions. Our focus is proving the unique monotone factorization theorem, which states that for a given permutation, there is only one monotone factorization of minimal length. To prove this, we employ inductive arguments on the structure of the Cayley graph. The key insight is using the connectivity of the graph and properties of shortest paths to characterize the uniqueness of the minimal factorization. This inductive approach allows us to rigorously connect the combinatorial Hurwitz problem to foundational graph concepts. Overall, this paper makes important theoretical advances in enumerating Hurwitz numbers by using Cayley graphs and induction to prove the novel unique monotone factorization theorem. The connections drawn between combinatorics, graphs, and inductive proofs are technically innovative. This theoretical foundation will hopefully stimulate further research into the deep links between the Hurwitz problem and other branches of mathematics.

A Graph Theorist Plants a Tree

Summary A graph theorist, while planting a tree, is inspired to study coloring problems for a class of graphs. Namely, she completely determines the chromatic number for certain Cayley graphs associated to the cross product of the integers with itself finitely many times, modulo a cyclic subgroup. This result she dubs the “Tree Guard Theorem.” She finds examples “in nature” of this theorem in action, and then seeds are sown for new directions to pursue.

Quantum circuit model for discrete-time three-state quantum walks on Cayley graphs

Quantum circuit model for discrete-time three-state quantum walks on Cayley graphs

Hyperfiniteness of boundary actions of relatively hyperbolic groups

We show that if G is a finitely generated group hyperbolic relative to a finite collection of subgroups \mathcal{P} , then the natural action of G on the geodesic boundary of the associated relative Cayley graph induces a hyperfinite equivalence relation. As a corollary of this, we obtain that the natural action of G on its Bowditch boundary \partial (G,\mathcal{P}) also induces a hyperfinite equivalence relation. This strengthens a result of Ozawa obtained for \mathcal{P} consisting of amenable subgroups and uses a recent work of Marquis and Sabok.

Universal inequalities for Dirichlet eigenvalues on discrete groups

We prove universal inequalities for Laplacian eigenvalues with Dirichlet boundary condition on subsets of certain discrete groups. The study of universal inequalities on Riemannian manifolds was initiated by Weyl, Pólya, Yau, and others. Here we focus on a version by Cheng and Yang. Specifically, we prove Yang-type universal inequalities for Cayley graphs of finitely generated amenable groups, as well as for the d -regular tree (simple random walk on the free group).

On an infinite family of integral Cayley graphs of Pauli groups

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 Cay(Pn,SPn) of the generalized Pauli group Pn on n-qubits; in fact, Pn may be decomposed as the central product of finite 2-groups, and a suitable choice of the generating set SPn allows us to recognize the structure of central product of graphs in Cay(Pn,SPn).Using this approach, we are able to recursively construct the adjacency matrix of Cay(Pn,SPn) for each n≥1, and to explicitly describe its spectrum and the associated eigenvectors. It turns out that Cay(Pn,SPn) is a (3n+2)-regular bipartite graph on 4n+1 vertices, and it has integral spectrum. This is a highly nontrivial property if one considers that, by choosing as a generating set for P1 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.