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...

Given a directed graph G, the K node-disjoint paths problem consists in finding a partition of G into K node-disjoint paths, such that each path ends up in a given subset of nodes in G. This article provides a necessary condition for the K node-disjoint paths problem which combines (1) the structure of the reduced graph associated with G, (2) the structure of each strongly connected component o...

The notion of a C-ultrahomogeneous graph, due to Isaksen et al., is adapted for digraphs, and then a strongly connected ~ C4-ultrahomogeneous oriented graph on 168 vertices and 126 pairwise arc-disjoint 4-cycles is presented, with regular indegree and outdegree 3 and no circuits of lengths 2 and 3, by altering a definition of the Coxeter graph via pencils of ordered lines of the Fano plane in w...

Forty years ago, Goethals and Seidel showed that if the adjacency algebra of a strongly regular graph X contains a Hadamard matrix then X is of Latin square type or of negative Latin square type [8]. We extend their result to complex Hadamard matrices and find only three additional families of parameters for which the strongly regular graphs have complex Hadamard matrices in their adjacency alg...