In a mixed (Δ, d)-regular graph, every vertex is incident with Δ ≥ 1 undirected edges and there are d ≥ 1 directed edges entering and leaving each vertex. If such a mixed graph has diameter 2, then its order cannot exceed (Δ+ d) + d+1. This quantity generalizes the Moore bounds for diameter 2 in the case of undirected graphs (when d = 0) and digraphs (when Δ = 0). For every d such that d − 1 is...

During a discussion taking place at WMC’01, G. Paun asked the question of what can be computed only by moving symbols between membranes. In this paper we provide some elements of the answer, in a setting similar of tissue P systems, where the set of membranes is organized as a Cayley graph and using a very simple propagation process characterizing accretive growth. Our main result is to charact...

We determine all periodic (and, therefore, all finite) semigroups G for which there exists a non-empty subset S of G such that the Cayley graph of G relative to S is an undirected Cayley graph. Let G be a semigroup, and let S be a nonempty subset of G. The Cayley graph Cay(G,S) of G relative to S is defined as the graph with vertex set G and edge set E(S) consisting of those ordered pairs (x, y...

A Cayley (resp. bi-Cayley) graph on a dihedral group is called dihedrant bi-dihedrant). In 2000, classification of trivalent arc-transitive dihedrants was given by Marušič and Pisanski, several years later, non-arc-transitive order 4p or 8p (p prime) were classified Feng et al. As generalization these results, our first result presents dihedrants. Using this, complete vertex-transitive non-Cayl...

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 ...

Zhou, H., The chromatic difference sequence of the Cartesian product of graphs: Part II, Discrete Applied Mathematics 41 (1993) 263-267. This paper is a continuation of our earlier paper under the same title. We prove that the normalized chromatic difference sequences of the Cartesian powers of a Cayley graph on a finite Abelian group are stable; and that if the chromatic difference sequence of...

We study the automorphisms of a Cayley graph that preserve its natural edge-colouring. More precisely, we are interested in groups G, such that every such automorphism of every connected Cayley graph on G has a very simple form: the composition of a left-translation and a group automorphism. We find classes of groups that have this property, and we determine the orders of all groups that do not...

In a recent paper Gunnells, Scott and Walden have determined the complete spectrum of the Schreier graph on the symmetric group corresponding to the Young subgroup Sn−2 × S2 and generated by initial reversals. In particular they find that the first nonzero eigenvalue, or spectral gap, of the Laplacian is always 1, and report that “empirical evidence” suggests that this also holds for the corres...

In this article current directions in solving Lovász’s problem about the existence of Hamilton cycles and paths in connected vertex-transitive graphs are given. © 2009 Elsevier B.V. All rights reserved. 1. Historical motivation In 1969, Lovász [59] asked whether every finite connected vertex-transitive graph has a Hamilton path, that is, a simple path going through all vertices, thus tying toge...

Roundness of metric spaces was introduced by Per Enflo as a tool to study uniform structures of linear topological spaces. The present paper investigates geometric and topological properties detected by the roundness of general metric spaces. In particular, we show that geodesic spaces of roundness 2 are contractible, and that a compact Riemannian manifold with roundness > 1 must be simply conn...