A spread of a strongly regular graph is a partition of the vertex set into cliques that meet Delsarte's bound (also called Hoffman's bound). Such spreads give rise to colorings meeting Hoffman's lower bound for the chromatic number and to certain imprimitive three-class association schemes. These correspondences lead to conditions for existence. Most examples come from spreads and fans in (part...

We consider strongly regular graphs r = (V, E) on an even number, say 2n, of vertices which admit an automorphism group G of order n which has two orbits on V. Such graphs will be called strongly regular semi-Cayley graphs. For instance, the Petersen graph, the Hoffman-Singleton graph, and the triangular graphs T(q) with q = 5 mod 8 provide examples which cannot be obtained as Cayley graphs. We...

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.

We use nonstandard topology and the g-topology to characterize normal, almost-normal, regular, almost-regular, semiregular spaces. The cluster [resp. 0-cluster] set relation is used to characterize regular, almostregular [resp. strongly-regular] spaces. The Whyburn [resp. Dickman] filter bases are characterized and it is shown that the cluster [resp. f-cluster] set relation restricted to the do...

In this paper, we examine the structure of vertexand edge-transitive strongly regular graphs, using normal quotient reduction. We show that the irreducible graphs in this family have quasiprimitive automorphism groups, and prove (using the Classification of Finite Simple Groups) that no graph in this family has a holomorphic simple automorphism group. We also find some constraints on the parame...

Using results on Hadamard difference sets, we construct regular graphical Hadamard matrices of negative type of order 4m for every positive integer m. If m > 1, such a Hadamard matrix is equivalent to a strongly regular graph with parameters (4m, 2m +m, m +m, m +m). Strongly regular graphs with these parameters have been called max energy graphs, because they have maximal energy (as defined by ...

We give characterizations of strong determinism for regular expressions with counting, based on which we present an O(|ΣE ||E|) time algorithm to check whether an expression E with counting is strongly deterministic where ΣE is the set of distinct symbols in E. It improves the previous upper bound of O(|E|) time on the same decision problems for both standard regular expressions and regular exp...

We show that if E is an arbitrary acyclic graph then the Leavitt path algebra LK(E) is locally K-matricial; that is, LK(E) is the direct union of subalgebras, each isomorphic to a finite direct sum of finite matrix rings over the fieldK. As a consequence we get our main result, in which we show that the following conditions are equivalent for an arbitrary graph E: (1) LK (E) is von Neumann regu...