As the main result of this paper it is proved that an interval matrix [Ac −∆, Ac +∆] is strongly regular if and only if the matrix inequality M(I − |I − RAc| − |R|∆) ≥ I has a solution, where M and R are real square matrices and M is nonnegative. Several consequences of this result are drawn.

We consider directed strongly regular graphs de2ned in 1988 by Duval. All such graphs with n vertices, n6 20, having a vertex-transitive automorphism group, are determined with the aid of a computer. As a consequence, we prove the existence of directed strongly regular graphs for three feasible parameter sets listed by Duval. For one parameter set a computer-free proof of the nonexistence is pr...

The action of PGL(2,2m) on the set of exterior lines to a nonsingular conic in PG(2,2m) affords an association scheme, which was shown to be pseudocyclic in [H.D.L. Hollmann, Association schemes, Master thesis, Eindhoven University of Technology, 1982]. It was further conjectured in [H.D.L. Hollmann, Association schemes, Master thesis, Eindhoven University of Technology, 1982] that the orbital ...

In this article we construct a series of new infinite families of strongly regular graphs with the same parameters as the point-graphs of non-singular quadrics in PG(n, 2). We study these graphs, describing and counting their maximal cliques, and determining their automorphism groups.

Suppose that n is even and a set of n 2 − 1 mutually orthogonal Latin squares of order n exists. Then we can construct a strongly regular graph with parameters (n, n 2 (n−1), n 2 ( 2 −1), n 2 ( 2 −1)), which is called a Latin square graph. In this paper, we give a sufficient condition of the Latin square graph for the existence of a projective plane of order n. For the existence of a Latin squa...

A partial difference set having parameters (n2, r(n − 1), n + r2 − 3r, r2 − r) is called a Latin square type partial difference set, while a partial difference set having parameters (n2, r(n + 1),−n + r2 + 3r, r2 + r) is called a negative Latin square type partial difference set. In this paper, we generalize well-known negative Latin square type partial difference sets derived from the theory o...

We give a sufficient condition for a distance-regular graph to be Hamiltonian. In particular, the Petersen graph is the only connected nonHamiltonian strongly regular graph on fewer than 99 vertices.

Strongly regular graphs lie on the cusp between highly structured and unstructured. For example, there is a unique strongly regular graph with parameters (36,10,4,2), but there are 32548 non-isomorphic graphs with parameters (36,15,6,6). (The first assertion is a special case of a theorem of Shrikhande, while the second is the result of a computer search by McKay and Spence.) In the light of th...

We survey recent results on constructions of difference sets and strongly regular Cayley graphs by using union of cyclotomic classes of finite fields. Several open problems are raised.

(otherwise we divide the vector by an appropriate scalar), so w.l.o.g. we have uj = 1 for a certain j ∈ {1, . . . , v}. The absolute value |(A~u)j| of the j-th component of A~u is at most ∑ i∼j |ui|; since the absolute values of all components of ~u are less than or equal to 1, we have ∑ i∼j |ui| ≤ k. On the other hand |(A~u)j| must be equal to |ρuj| = |ρ|, from which we obtain |ρ| ≤ k. If ρ = ...