We prove two results for directed strongly regular graphs that have an eigenvalue of multiplicity less than k, the common out-degree of each vertex. The first bounds the size of an independent set, and the second determines an eigenvalue of the subgraph on the out-neighborhood of a vertex. Both lead to new nonexistence results for parameter sets.

In this paper we characterize the automorphisms (collineations) of the Segre product of partial linear spaces. In particular, we show that if all components of the product are strongly connected, then every collineation is determined by a set of isomorphisms between its components. The formalization follows the ideas presented in the Journal of Geometry paper [16] by Naumowicz and Prażmowski.

We introduce canonical principal parameters on any strongly regular minimal surface in the three dimensional sphere and prove that any such a surface is determined up to a motion by its normal curvature function satisfying the Sinh-Poisson equation. We obtain a classification theorem for bi-umbilical hypersurfaces of type number two. We prove that any minimal hypersurface of type number two wit...

We prove cocompleteness of the category of CASL signatures , of monotone signatures, of strongly regular signatures and of strongly locally ltered signatures. This shows that using these signature categories is compatible with a pushout or colimit based module system.

The chromatic number of a Latin square is the least number of partial transversals which cover its cells. This is just the chromatic number of its associated Latin square graph. Although Latin square graphs have been widely studied as strongly regular graphs, their chromatic numbers appear to be unexplored. We determine the chromatic number of a circulant Latin square, and find bounds for some ...

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...

Let (B t (s); 0 s < 1) be reeecting inhomogeneous Brownian motion with drift t ? s at time s, started with B t (0) = 0. Consider the random graph G(n; n ?1 +tn ?4=3), whose largest components have size of order n 2=3. Normalizing by n ?2=3 , the asymptotic joint distribution of component sizes is the same as the joint distribution of excursion lengths of B t (Corollary 2). The dynamics of mergi...

In this paper we examine some natural ideal conditions and show how graphs can be defined that give a visualization of these conditions. We the interplay between multiplicative theory graph theoretic structure associated graph. Behavior under standard ring extensions are studied, in conjunction with theory, classical results connections made.

The general symplectic graph Sp(2ν, q) is introduced. It is shown that Sp(2ν, q) is strongly regular. Its parameters are computed, its chromatic number and group of graph automorphisms are also determined.