A number of researchers have proposed Cayley graphs and Schreier coset graphs as models for interconnection networks. New algorithms are presented for generating Cayley graphs in a more time-eecient manner than was previously possible. Alternatively, a second algorithm is provided for storing Cayley graphs in a space-eecient manner (log 2 (3) bits per node), so that copies could be cheaply stor...

The well-known conjecture that there are no snarks amongst Cayley graphs is considered. Combining the theory of Cayley maps with the existence of certain kinds of independent sets of vertices in arc-transitive graphs, some new partial results are obtained suggesting promising future research directions in regards to this conjecture.

We describe a method for nding a minimal presentation of a nite group with the help of rewrite techniques. We use the correspondence between group relators and circuits in Cayley graphs to deene speciic inference rules, speeding up the completion procedure. Our framework is aimed at solving shortest path routing problem in Cayley graphs.

We characterize the set of planar locally finite Cayley graphs, and give a finite representation of these graphs by a special kind of finite state automata called labeling schemes. As a result, we are able to enumerate and describe all planar locally finite Cayley graphs of a given degree. This analysis allows us to solve the problem of decision of the locally finite planarity for a word-proble...

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.

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.

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.

For any d ≥ 5 and k ≥ 3 we construct a family of Cayley graphs of degree d, diameter k, and order at least k((d−3)/3)k. By comparison with other available results in this area we show that, for all sufficiently large d and k, our family gives the current largest known Cayley graphs of degree d and diameter k.

We show that Cayley graphs of finitely generated Abelian groups are rather rigid. As a consequence we obtain that two finitely generated Abelian groups admit isomorphic Cayley graphs if and only if they have the same rank and their torsion parts have the same cardinality. The proof uses only elementary arguments and is formulated in a geometric language.

Self-complementary Cayley graphs are useful in the study of Ramsey numbers, but they are relatively very rare and hard to construct. In this paper, we construct several families of new self-complementary Cayley graphs of order p4 where p is a prime and congruent to 1 modulo 8.