A colouring of a strongly regular graph is an allocation of colours (or treatments) to the vertices of the graph. Such a colouring is balanced if every pair of distinct colours occurs equally often on the ends of an edge. When the graph is the complete regular multipartite graph a balanced colouring is just a balanced incomplete-block design, or 2-design. Some constructions are given. For examp...

A graph Γ with diameter d is strongly distance-regular if Γ is distanceregular and its distance-d graph Γd is strongly regular. The known examples are all the connected strongly regular graphs (i.e. d = 2), all the antipodal distanceregular graphs, and some distance-regular graphs with diameter d = 3. The main result in this paper is a characterization of these graphs (among regular graphs with...

the concept of algebraic hyperstructures introduced by marty as a generalization of ordinary algebraic structures. in an ordinary algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. the concept of ?-semihyperrings is a generalization of semirings, a generalization of semihyper rings and a generalizat...

In this paper, we study a conjecture of Andries E. Brouwer from 1996 regarding the minimum number of vertices of a strongly regular graph whose removal disconnects the graph into non-singleton components. We show that strongly regular graphs constructed from copolar spaces and from the more general spaces called ∆-spaces are counterexamples to Brouwer’s Conjecture. Using J.I. Hall’s characteriz...

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...

A graph G of even order v is called t-extendable if it contains a perfect matching, t < v/2 and any matching of t edges is contained in some perfect matching. The extendability of G is the maximum t such that G is t-extendable. In this paper, we study the extendability properties of strongly regular graphs. We improve previous results and classify all strongly regular graphs that are not 3-exte...