A graph is 1-arc-regular if its full automorphism group acts regularly on the set of its arcs. In this paper, we investigate 1-arc-regular graphs of prime valency, especially of valency 3. First, we prove that if G is a soluble group then a (G, 1)-arc-regular graph must be a Cayley graph of a subgroup of G . Next we consider trivalent Cayley graphs of a finite nonabelian simple group and obtain...

This thesis investigates uniform mixing on Cayley graphs over Z3. We apply Mullin’s results on Hamming quotients, and characterize the 2(d+2)-regular connected Cayley graphs over Z3 that admit uniform mixing at time 2π/9. We generalize Chan’s construction on the Hamming scheme H(d, 2) to the schemeH(d, 3), and find some distance graphs of the Hamming graph H(d, 3) that admit uniform mixing at t...

Skoviera, M. and J. ‘&Iii, Regular maps from Cayley graphs, Part i: Balanced Cayley maps, Discrete Mathematics IO9 (1992) 265-276. A Cayley map is a Cayley graph 2-cell embedded in some orientable surface so that the local rotations at every vertex are identical. Two types of Cayley maps are introduced: the balanced and antibalanced Cayley maps. In Part 1, conditions are given uuder which a bal...

We introduce a new family of interconnection networks that are Cayley graphs with xed degrees of any even number greater than or equal to 4. We call the proposed graphs cyclic-cubes because contracting some cycles in such a graph results in a generalized hypercube. These Cayley graphs have optimal fault tolerance and logarithmic diameters. For comparable number of nodes, a cyclic-cube can have ...

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...