The symmetry properties of mathematical structures are often important for understanding these structures. Graphs sometimes have a large group of symmetries, especially when they have an algebraic construction such as the Cayley graphs. These graphs are constructed from abstract groups and are vertex-transitive and this is the reason for their symmetric appearance. Some Cayley graphs have even ...

Let R be a finite commutative ring. The unitary Cayley graph of R, denoted GR, is the graph with vertex set R and edge set {{a, b} : a, b ∈ R, a− b ∈ R×}, where R× is the set of units of R. An r-regular graph is Ramanujan if the absolute value of every eigenvalue of it other than ±r is at most 2 √ r − 1. In this paper we give a necessary and sufficient condition for GR to be Ramanujan, and a ne...

Let [Formula: see text] be a commutative ring and the multiplicative group of unit elements text]. In 2012, Khashyarmanesh et al. defined generalized unitary Cayley graph, text], corresponding to subgroup nonempty subset with as graph vertex set text]and two distinct vertices being adjacent if only there exists such that this paper, we characterize all Artinian rings for which is projective. Th...

A graph $Gamma$ is said to be vertex-transitive or edge‎- ‎transitive‎ ‎if the automorphism group of $Gamma$ acts transitively on $V(Gamma)$ or $E(Gamma)$‎, ‎respectively‎. ‎Let $Gamma=Cay(G,S)$ be a Cayley graph on $G$ relative to $S$‎. ‎Then, $Gamma$ is said to be normal edge-transitive‎, ‎if $N_{Aut(Gamma)}(G)$ acts transitively on edges‎. ‎In this paper‎, ‎the eigenvalues of normal edge-tra...

 In this paper, we introduce a suitable generalization of Cayley graphs that is defined over polygroups (GCP-graph) and give some examples and properties. Then, we mention a generalization of NEPS that contains some known graph operations and apply to GCP-graphs. Finally, we prove that the product of GCP-graphs is again a GCP-graph.

Let Cay(H, S), S/H be an arbitrary Cayley graph over a finite group H. We shall say that two Cayley graphs Cay(H, S) and Cay(H, T) are Cayley isomorphic if there exists an automorphism . # Aut(H ) such that S=T. It is a trivial observation that two Cayley isomorphic Cayley graphs are isomorphic as graphs. The converse is not true: two Cayley graphs over the same group may be isomorphic as graph...

for two normal edge-transitive cayley graphs on groups h and k which have no common direct factor and gcd(jh=h ′j; jz(k)j) = 1 = gcd(jk=k ′j; jz(h)j), we consider four standard products of them and it is proved that only tensor product of factors can be normal edge-transitive.