Spreads of finite symplectic, orthogonal and unitary vector spaces are used to construct new strongly regular graphs having the same parameters as the perpendicularity graphs of the underlying vector spaces. Some of the graphs are. related to partial geometries, while others produce interesting symmetric designs.

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...

In this paper we construct transitive t-designs from the linear groups L(2, q), q ≤ 23. Thereby we classify t-designs, t ≥ 2, admitting a transitive action of the linear groups L(2, q), q ≤ 23, up to 35 points and obtained numerous transitive designs, for 36 ≤ v ≤ 55. In many cases we proved the existence of t-designs with certain parameter sets. Among others we constructed t-designs with param...

A graph is called of type k if it is connected, regular, and has k distinct eigenvalues. For example graphs of type 2 are the complete graphs, while those of type 3 are the strongly regular graphs. We prove that for any positive integer n, every graph can be embedded in n cospectral, non-isomorphic graphs of type k for every k ≥ 3. Furthermore, in the case k ≥ 5 such a family of extensions can ...

In this paper, by using minimum out-degree and minimum in-degree, we give a new lower bound on the vertex-strong connectivity of an oriented graph. In the case of a tournament, our lower bound improves that of Thomassen obtained in 1980 and which use the notion of irregularity (see [C. Thomassen, Hamiltonian-connected tournaments, J. Combin. Theory Ser. B 28 (1980) 142–163]). As application, we...

We discuss relations between several known (some false, some open) conjectures on 3-edgeconnected, cubic graphs and how they all fit into the same framework related to cuts in matchings. We then provide a construction of 3-edge-connected digraphs satisfying the property that for every even subgraph E, the graph obtained by contracting the edges of E is not strongly connected. This disproves a r...

Let G be a strongly connected, weighted directed graph. We define a product distance η(i, j) for pairs i, j of vertices and form the corresponding product distance matrix. We obtain a formula for the determinant and the inverse of the product distance matrix. The edge orientation matrix of a directed tree is defined and a formula for its determinant and its inverse, when it exists, is obtained....

Combining results on quadrics in projective geometries with an algebraic interplay between finite fields and Galois rings, we construct the first known family of partial difference sets with negative Latin square type parameters in nonelementary abelian groups, the groups Z 4 × Z4`−4k 2 for all k when ` is odd and for all k < ` when ` is even. Similarly, we construct partial difference sets wit...

The cell rotation graph D(G) on the strongly connected orientations of a 2-edge-connected plane graph G is de5ned. It is shown that D(G) is a directed forest and every component is an in-tree with one root; if T is a component of D(G), the reversions of all orientations in T induce a component of D(G), denoted by T−, thus (T; T−) is called a pair of in-trees of D(G); G is Eulerian if and only i...

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...