Strongly regular Cayley graphs with Paley parameters over abelian groups of rank 2 were studied in [5] and [12]. It was shown that such graphs exist iff the corresponding group is isomorphic to Zpn ⊕Zpn , where p is an odd prime. In this paper we classify all strongly regular Cayley graphs over this group using Schur rings method. As a consequence we obtain a complete classification of strongly...

The distance d(f, f ′) between two triangular embeddings f and f ′ of a complete graph is the minimal number t such that we can replace t faces in f by t new faces to obtain a triangular embedding isomorphic to f ′. We consider the problem of determining the maximum value of d(f, f ′) as f and f ′ range over all triangular embeddings of a complete graph. The following theorem is proved: for eve...

A basic problem in the design of mobile telephone networks is to assign sets of radio frequency bands (colours) to transmitters (vertices) to avoid interference. Often the transmitters are laid out like vertices of a triangular lattice in the plane. We investigate the corresponding colouring problem of assigning sets of colours of size p(v) to each vertex of the triangular lattice so that the s...

We present a Prym construction which associates abelian varieties to vertex-transitive strongly regular graphs. As an application we construct Prym-Tyurin varieties of arbitrary exponent ≥ 3, generalizing a result by Lange, Recillas and Rochas.

ON GRAPHS SATISFYING A STRONG ADJACENCY PROPERTY Y. Ananchuen and L. Caccetta School of Mathematics and Statistics Curtin University of Technology GPO Box U1987 Perth 6001 Western Australia. Dedicated to the memory of Alan Rahilly, 1947 1992 Let m and n be nonnegative integers and k be a positive integer. A graph G is said to have property P*(m,n,k) if for any set of m + n distinct vertices of ...

Dedicated to the memory of Prof. Dom de Caen Abstract. Given λ and μ, it is shown that, unless one of the relations (λ− μ)2 = 4μ, λ− μ = −2, (λ− μ)2 + 2(λ− μ) = 4μ is satisfied, there are only finitely many parameter sets (v, k, λ, μ) for which a strongly regular graph with these parameters could exist. This result is used to prove as special cases a number of results which have appeared in pre...

In a previous paper an incomplete investigation into regular two-graphs on 36 vertices established the existence of at least 227. Using a more efficient algorithm, the two authors have independently verified that in fact these 227 comprise the complete set. An immediate consequence of this is that all strongly regular graphs with parameters (35,16,6,8), (36,14,4,6), (36,20,10,12) and their comp...

We consider strongly regular graphs defined on a finite field by taking the union of some cyclotomic classes as difference set. Several new examples are found.

Directed graphs are widely used in modelling of nonsymmetric relations in various sciences and engineering disciplines. We discuss invariants of strongly connected directed graphs minimal number of vertices or edges necessary to remove to make remaining graphs not strongly connected. By analogy with undirected graphs these invariants are called strong vertex/edge connectivities. We review first...

Considering the Euclidean Jordan algebra of the real symmetric matrices endowed with the Jordan product and the inner product given by the usual trace of matrices, we construct an alternating Schur series with an element of the Jordan frame associated to the adjacency matrix of a strongly regular graph. From this series we establish necessary conditions for the existence of strongly regular gra...