Relations between Laplacian eigenvectors and eigenvalues and the existence of almost equitable partitions (which are generalizations of equitable partitions) are presented. Furthermore, on the basis of some properties of the adjacency eigenvectors of a graph, a necessary and sufficient condition for the graph to be primitive strongly regular is introduced. c © 2006 Elsevier Ltd. All rights rese...

E. R. van Dam gave an example of primitive non-amorphous association schemes in which every nontrivial relation is a strongly regular graph, as a fusion scheme of the cyclotomic scheme of class 45 on GF(212). The aim of this paper is to present a new example of primitive non-amorphous association schemes in which every nontrivial relation is a strongly regular graph, as a fusion scheme of the c...

Almost all finite graphs have the n-e.c. adjacency property, although until recently few explicit examples of such graphs were known. We survey some recently discovered families of explicit finite n-e.c. graphs, and present a new construction of strongly regular n-e.c. arising from affine planes.

We give an alternative proof of the fact that a planar function cannot exist on groups of even order. The argument involved leads us to define a class of functions which we call semi-planar. Through the introduction of an incidence structure we construct semi-biplanes using semi-planar functions. The method involved represents a new approach to constructing semi-biplanes and provides infinite c...

We prove that any strongly regular Weingarten surface in Euclidean space carries locally geometric principal parameters. The basic theorem states that any strongly regular Weingarten surface is determined up to a motion by its structural functions and the normal curvature function satisfying a geometric differential equation. We apply these results to the special Weingarten surfaces: minimal su...

This note presents an implicit function theorem for generalized equations, simultaneously generalizing Robinson’s implicit function theorem for strongly regular generalized equations and Clarke’s implicit function theorem for equations with Lipschitz-continuous mappings.

The purpose of this paper is to define a new equivalence relation τ∗ on divisible hypergroups and to show that this relation is the smallest strongly regular relation (the fundamental relation) on commutative divisible hypergroups. We show that τ∗ ̸= β∗, τ∗ ̸= γ∗ and, we define a divisible hypergroup on every nonempty set. We show that the quotient of a finite divisible hypergroup by τ∗ is the tr...

We provide a new construction of the strongly regular graphs associated with the two sporadic simple groups M22 and HS. Further, we give some new constructions of other known strongly regular graphs by taking the orbits of a certain subgroup of M22 on the planes of the Hermitian variety H(5, 4). These geometric constructions can be used to produce cap codes with large parameters and automorphis...

For every n positive integer we show the construction of a strongly regular graph of order at most 2 which contains every graph of ordered n as a subgraph. The estimation concerning the construction is best possible.

In a previous paper it was established that there are at least 27 non-isomorphic strongly regular (40, 12, 2, 4) graphs. Using a different and more efficient method we have re-investigated these graphs and have now been able to determine them all, and so complete the classification. We have discovered that there are precisely 28 non-isomorphic (40, 12, 2, 4) strongly regular graphs. The one tha...