Related papers: The generalized X-join of Cayley graphs
We introduce a family of graphs that generalises the class of Cayley graphs. For non-empty subsets L, R of a group G, the two-sided Cayley graph 2SC(G;L,R) is the directed graph with vertex set G and an arc from x to y if and only if…
Isomorphic factorizations of complete graphs originate from the seminal work of Frank Harary and collaborators, who initiated the systematic study of decompositions of complete graphs into pairwise isomorphic spanning subgraphs. In this…
Cayley graphs are graphs on algebraic structures, typically groups or group-like structures. In this paper, we have obtained a few results on Cayley graphs on Cyclic groups, powers of cycles, Cayley graphs on some non-abelian groups, and…
Suppose $X$ is a simple graph. The $X-$join $\Gamma$ of a set of complete or empty graphs $\{X_x \}_{x \in V(X)}$ is a simple graph with the following vertex and edge sets: \begin{eqnarray*} V(\Gamma) &=& \{(x,y) \ | \ x \in V(X) \ \& \ y…
Let $S$ be a set of transpositions such that the girth of the transposition graph of $S$ is at least 5. It is shown that the automorphism group of the Cayley graph of the permutation group $H$ generated by $S$ is the semidirect product…
In this work, we try to enunciate the Total chromatic number of some Cayley graphs like the Cayley graph on Symmetric group, Alternating group, Dihedral group with respect to some generating sets and some other regular graphs.
For a transitive infinite connected graph $G$, let $\mu(G)$ be its connective constant. Denote by $\mathbf{\cal G}$ the set of Cayley graphs for finitely generated infinite groups with an infinite-order generator which is independent of…
The relative Cayley graph of a group $G$ with respect to its proper subgroup $H$, is a graph whose vertices are elements of $G$ and two vertices $h\in H$ and $g\in G$ are adjacent if $g=hc$ for some $c\in C$, where $C$ is an inversed-closed…
We develop a theory of soficity for actions on graphs and obtain new applications to the study of sofic groups. We establish various examples, stability and permanence properties of sofic actions on graphs, in particular soficity is…
We present simple graph-theoretic characterizations of Cayley graphs for monoids, semigroups and groups. We extend these characterizations to commutative monoids, semilattices, and abelian groups.
This paper deals with some of the algebraic properties of Sierpi\'nski graphs and a family of regular generalized Sierpi\'nski graphs. For the family of regular generalized Sierpi\'nski graphs, we obtain their spectrum and characterize…
When one studies geometric properties of graphs, local finiteness is a common implicit assumption, and that of transitivity a frequent explicit one. By compactness arguments, local finiteness guarantees several regularity properties. It is…
A Cayley Graph for a group $G$ is called normal edge-transitive if it admits an edge-transitive action of some subgroup of the Holomorph of $G$ (the normaliser of a regular copy of $G$ in $\operatorname{Sym}(G)$). We complete the…
Let $X$ be a connected Cayley graph on an abelian group of odd order, such that no two distinct vertices of $X$ have exactly the same neighbours. We show that the direct product $X \times K_2$ (also called the "canonical double cover" of…
New criteria for which Cayley graphs of cyclic groups of any order can be completely determined--up to isomorphism--by the eigenvalues of their adjacency matrices is presented. Secondly, a new construction for pairs of nonisomorphic Cayley…
Let $X=GD$ be a group, where $G$ is a nonabelian simple group and $D$ is a dihedral group. These groups $X$ are closely related to regular Cayley maps. The main theorems of this paper describes $X$.
We study subsets $T$ consisting of some transpositions $(i,j)$ of the symmetric group $S_n$ on $\{1,\dots,n\}$ such that the Cayley graph $\Gamma_T:=Cay(S_n,T)$ is an integral graph, i.e., all eigenvalues of an adjacency matrix of…
Recently, several works by a number of authors have provided characterizations of integral undirected Cayley graphs over generalized dihedral groups and generalized dicyclic groups. We generalize and unify these results in two different…
Given the Cayley graph of a finitely generated group $G$, with respect to a presentation $G^{\alpha}$ with $n$ generators, the quotient of the rank of the fundamental group of subgraphs of the Cayley graph by the cardinality of the set of…
Tournaments are graphs obtained by assigning a direction for every edge in an undirected complete graph. We give a formula for the number of isomorphism classes of vertex-transitive tournaments with prime order. For that, we introduce…