We consider strongly regular graphs defined on a finite field by taking the union of some cyclotomic classes as difference set. Several new examples are found.

Considering the Euclidean Jordan algebra of the real symmetric matrices endowed with the Jordan product and the inner product given by the usual trace of matrices, we construct an alternating Schur series with an element of the Jordan frame associated to the adjacency matrix of a strongly regular graph. From this series we establish necessary conditions for the existence of strongly regular gra...

A new family of distance-regular graphs is constructed. They are antipodal 2 -fold covers of the complete graph on 2 vertices. The automorphism groups are determined, and the extended Preparata codes are reconstructed using walks on these graphs. There are connections to other interesting structures: the graphs are equivalent to certain generalized Hadamard matrices; and their underlying 3-clas...

In this paper, we prove the following two theorems. Theorem 1 Let 0 denote a distance-regular graph with diameter d ≥ 3. Suppose E and F are primitive idempotents of 0, with cosine sequences σ0, σ1, . . . , σd and ρ0, ρ1, . . . , ρd , respectively. Then the following are equivalent. (i) The entry-wise product E ◦ F is a scalar multiple of a primitive idempotent of 0. (ii) There exists a real nu...

Let q be a power of a prime and E be an elliptic curve defined over Fq. In [17], the present author examined a sequence of polynomials which express the Nk’s, the number of points on E over the field extensions Fqk , in terms of the parameters q and N1 = #E(Fq). These polynomials have integral coefficients which alternate in sign, and a combinatorial interpretation in terms of spanning trees of...

We show that the contact process on a random d-regular graph initiated by a single infected vertex obeys the “cutoff phenomenon” in its supercritical phase. In particular, we prove that, when the infection rate is larger than the lower critical value of the contact process on the infinite d-regular tree, there are positive constants C, p depending on the infection rate such that for any ε > 0, ...

A distinction is made between data description and representational space in the context of circumplex models. The representational space provides the language in which data are described, and different languages have their advantages and disadvantages. For instance, points in a two-dimensional Cartesian grid can form a circle. Such a circular pattern corresponds to a description of the data pa...

A new family of distance-regular graphs is constructed. They are antipodal 22t−1-fold covers of the complete graph on 22t vertices. The automorphism groups are determined, and the extended Preparata codes are reconstructed using walks on these graphs. There are connections to other interesting structures: the graphs are equivalent to certain generalized Hadamard matrices; and their underlying 3...

Forty years ago, Goethals and Seidel showed that if the adjacency algebra of a strongly regular graph X contains a Hadamard matrix then X is of Latin square type or of negative Latin square type [8]. We extend their result to complex Hadamard matrices and find only three additional families of parameters for which the strongly regular graphs have complex Hadamard matrices in their adjacency alg...

Infinite families of linear binary nested completely regular codes with covering radius ρ equal to 3 and 4 are constructed. In the usual way, i.e., as coset graphs, infinite families of embedded distance-regular coset graphs of diameter D = 3 or 4 are constructed. In some cases, the constructed codes are also completely transitive codes and the corresponding coset graphs are distance-transitive.