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...

Given k ∈ N, the k’th discrete Heisenberg group, denoted H Z , is the group generated by the elements a1, b1, . . . , ak, bk, c, subject to the commutator relations [a1, b1] = . . . = [ak, bk] = c, while all the other pairs of elements from this generating set are required to commute, i.e., for every distinct i, j ∈ {1, . . . , k} we have [ai, aj ] = [bi, bj ] = [ai, bj ] = [ai, c] = [bi, c] = ...

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.

In this note we study embeddings of Cayley graphs of right groups on surfaces. We characterize those right groups which have a toroidal but no planar Cayley graph, such that the generating system of the right group has a minimal generating system of the group as a factor.

We consider contact processes on general Cayley graphs. It is shown that any such contact process has a well-defined exponential growth rate, which can be related to the configuration seen from a ‘typical’ infected site at a ‘typical’ late time. Using this quantity, it is proved that on any nonamenable Cayley graph, the critical contact process dies out.

The \arrowhead torus" is a broadcast graph that we deene on the 6-valent grid as a Cayley graph. We borrow the term from Mandelbrot who qualiies in that way one of the Sierpinski's famous fractal constructions. The 6-valent grid H = (V; E) is generated by three families of straight lines. We adopt the isotropic orientation S ! N, NE ! SW, NW ! SE and deene the system of generators S = fs 1 ; s ...