In this paper, a characterisation is given of finite s-arc transitive Cayley graphs with s ≥ 2. In particular, it is shown that, for any given integer k with k ≥ 3 and k 6= 7, there exists a finite set (maybe empty) of s-transitive Cayley graphs with s ∈ {3, 4, 5, 7} such that all s-transitive Cayley graphs of valency k are their normal covers. This indicates that s-arc transitive Cayley graphs...

Let G be a finite abelian group written additively with identity 0, and Ω an inverse closed generating subset of such that 0 ∉ Ω. We say has the property ‘‘us’’ (unique summation), whenever for every ≠ g ∈ if there are s1, s2, s3, s4 s1 + s2 = s3 s4, then we have {s1, s2} {s3, s4}. Cayley graph Γ Cay(G;Ω) is us-Cayley graph, ‘‘us’’. In this paper, show Aut(Γ) L(G) ⋊ A, where left regular repres...

The spined cube SQn, as a variant network of the hypercube Qn, was proposed in 2011 and has attracted much attention because its smaller diameter. It is well-known that Qn Cayley graph. In present paper, we show SQn an m-Cayley graph, automorphism group semiregular subgroup acting on vertices with m orbits, where m=4 when n≥6 m=⌊n/2⌋ n≤5. Consequently, it shows can be partitioned into eight dis...

We consider the lossless compression of vertex transitive graphs. An undirected graph G = (V, E) is called vertex transitive if for every pair of vertices x, y ∈ V , there is an automorphism σ of G, such that σ(x) = y. A result due to Sabidussi, guarantees that for every vertex transitive graph G there exists a graph mG (m is a positive integer) which is a Cayley graph. We propose as the compre...

In this paper, we introduce a class of double coset cayley digraphs induced by right solvable ward groupoids. These class of graphs can be viewed as a generalization of double coset cayley graphs induced by groups. Further, many graph properties are expressed in terms of algebraic properties.

A Cayley graph Γ = Cay(G, S) is called normal for G, if GR, the right regular representation of G, is a normal subgroup of the full automorphism group Aut(Γ ) of Γ . In this paper we determine the normality of connected and undirected Cayley graphs of valency three for dihedral groups. 2000 Mathematics Subject Classification: 05C25, 20B25.

We construct a polynomial-time algorithm that given a graph X with 4p vertices (p is prime), finds (if any) a Cayley representation of X over the group C2 × C2 × Cp. This result, together with the known similar result for circulant graphs, shows that recognising and testing isomorphism of Cayley graphs over an abelian group of order 4p can be done in polynomial time.

Preface The main goal of this book is to describe several tools of the quasi-isometric rigidity and to illustrate them by presenting (essentially self-contained) proofs of several fundamental theorems in this area: Gromov's theorem on groups of polynomial growth, Mostow Rigidity Theorem and Schwartz's quasi-isometric rigidity theorem for nonuniform lattices in the real-hyperbolic spaces. We con...

In this paper we introduce a new class of double coset Cayley digraphs induced by loops. These graphs can be considered as the generalization of Double coset Cayley digraphs induced by groups . Moreover, various graph properties are expressed in terms of algebraic properties. This did not attract much attention in the literature. Mathematics Subject Classification: 05C25

A graph is called integral, if its adjacency eigenvalues are integers. In this paper we determine integral quartic Cayley graphs on finite abelian groups. As a side result we show that there are exactly 27 connected integral Cayley graphs up to 11 vertices.