Let Γ = Cay(G, S) and G ≤ X ≤ AutΓ. We say Γ is (X, 1)-regular Cayley graph if X acts regularly on its arcs. Γ is said to be corefree if G is core-free in some X ≤ Aut(Cay(G, S)). In this paper, we prove that if an (X, 1)-regular Cayley graph of valency 5 is not normal or binormal, then it is the normal cover of one of two core-free ones up to isomorphism. In particular, there are no core-free ...

Hexagonal mesh and torus, as well as honeycomb and certain other pruned torus networks, are known to belong to the class of Cayley graphs which are node-symmetric and possess other interesting mathematical properties. In this paper, we use Cayley-graph formulations for the aforementioned networks, along with some of our previous results on subgraphs and coset graphs, to draw conclusions relatin...

Let G be a finite group, and let 1G 6∈ S ⊆ G. A Cayley di-graph Γ = Cay(G,S) of G relative to S is a di-graph with a vertex set G such that, for x, y ∈ G, the pair (x, y) is an arc if and only if yx−1 ∈ S. Further, if S = S−1 := {s−1|s ∈ S}, then Γ is undirected. Γ is conected if and only if G = 〈s〉. A Cayley (di)graph Γ = Cay(G,S) is called normal if the right regular representation of G is a ...

The connective constant μ(G) of an infinite transitive graph G is the exponential growth rate of the number of self-avoiding walks from a given origin. In earlier work of Grimmett and Li, a locality theorem was proved for connective constants, namely, that the connective constants of two graphs are close in value whenever the graphs agree on a large ball around the origin. A condition of the th...

We study some properties of the Cayley graph of the R.Thompson’s group F in generators x0, x1. We show that the density of this graph, that is, the least upper bound of the average vertex degree of its finite subgraphs is at least 3. It is known that a 2generated group is not amenable if and only if the density of the corresponding Cayley graph is strictly less than 4. It is well known this is ...

We characterize Cayley graphs of abelian groupswhich admit a nowhere-zero 3-flow. In particular, we prove that every k-valent Cayley graph of an abelian group, where k 4, admits a nowhere-zero

A digraph Γ is called n-Cayley digraph over a group G, if there exists a semiregular subgroup RG of Aut(Γ) isomorphic to G with n orbits. In this paper, we represent the adjacency matrix of Γ as a diagonal block matrix in terms of irreducible representations of G and determine its characteristic polynomial. As corollaries of this result we find: the spectrum of semi-Cayley graphs over abelian g...

The connective constant of a graph is the exponential growth rate of the number of self-avoiding walks starting at a given vertex. Strict inequalities are proved for connective constants of vertex-transitive graphs. First, the connective constant decreases strictly when the graph is replaced by a nontrivial quotient graph. Second, the connective constant increases strictly when a quasitransitiv...

A graph is vertex-transitive if its automorphism group acts transitively on its vertices. A vertex-transitive graph is a Cayley graph if its automorphism group contains a subgroup acting regularly on its vertices. In this paper, the cubic vertextransitive non-Cayley graphs of order 8p are classified for each prime p. It follows from this classification that there are two sporadic and two infini...

Cayley graph is a graph constructed out of a group Γ and a generating set A ⊆ Γ. When Γ = Zn, the corresponding Cayley graph is called as a circulant graph and denoted by Cir(n, A). In this paper, we attempt to find the total domination number of Cir(n, A) for a particular generating set A of Zn and a minimum total dominating set of Cir(n, A). Further, it is proved that Cir(n, A) is 2-total exc...