A graph is n-e.c. (n-existentially closed) if for every pair of subsets U , W of the vertex set V of the graph such that U ∩W = / 0 and |U |+ |W | = n, there is a vertex v ∈ V − (U ∪W ) such all edges between v and U are present and no edges between v and W are present. A graph is strongly regular if it is a regular graph such that the number of vertices mutually adjacent to a pair of vertices ...

We give a construction of strongly regular Cayley graphs on finite fields Fq by using union of cyclotomic classes and index 4 Gauss sums. In particular, we obtain two infinite families of strongly regular graphs with new parameters.

It is known that a strongly regular semi-Cayley graph (with respect to a group G) corresponds to a triple of subsets (C, D, D') of G. Such a triple (C, D, D') is called a partial difference triple. First, we study the case when D U D' is contained in a proper normal subgroup of G. We basically determine all possible partial difference triples in this case. In fact, when \G\ f 8 nor 25, all part...

Intriguing sets of vertices have been studied for several classes of strongly regular graphs. In the present paper, we study intriguing sets for the graphs Γn, n ≥ 2, which are defined as follows. Suppose Q(2n, 2), n ≥ 2, is a nonsingular parabolic quadric of PG(2n, 2) and Q+(2n − 1, 2) is a nonsingular hyperbolic quadric obtained by intersecting Q(2n, 2) with a suitable nontangent hyperplane. ...

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 give two constructions of strongly regular Cayley graphs on finite fields Fq by using union of cyclotomic classes and index 2 Gauss sums. In particular, we obtain twelve infinite families of strongly regular graphs with new parameters.

Let Γ be a distance-regular graph with diameter d and Kneser graph K = Γd, the distance-d graph of Γ. We say that Γ is partially antipodal when K has fewer distinct eigenvalues than Γ. In particular, this is the case of antipodal distance-regular graphs (K with only two distinct eigenvalues), and the so-called half-antipodal distance-regular graphs (K with only one negative eigenvalue). We prov...

Jacroux (1985) extended the definition of Regular graph designs of Mitchell and John (1976) to Semi-regular graph (SRG) designs, and studied the type 1 optimality of block designs. Here, the construction and optimality of some more SRG designs are discussed. Moreover, in this investigation, it is established that a class of two associate partially efficiency balanced design is, in fact SRG desi...

A graph G has constant μ = μ(G) if any two vertices that are not adjacent have μ common neighbours. G has constant μ and μ if G has constant μ = μ(G), and its complement G has constant μ = μ(G). If such a graph is regular, then it is strongly regular, otherwise precisely two vertex degrees occur. We shall prove that a graph has constant μ and μ if and only if it has two distinct restricted Lapl...