Generalized Krein Parameters of a Strongly Regular Graph

Lus Almeida Vieira,Vasco Moço Mano

doi:10.4236/am.2015.61005

Abstract

We consider the real three-dimensional Euclidean Jordan algebra associated to a strongly regular graph. Then, the Krein parameters of a strongly regular graph are generalized and some generalized Krein admissibility conditions are deduced. Furthermore, we establish some relations between the classical Krein parameters and the generalized Krein parameters.

Highlights

In this paper we explore the close and interesting relationship of a three-dimensional Euclidean Jordan algebra to the adjacency matrix of a strongly regular graph X
Euclidean Jordan algebras were born by adding an inner product with a certain property to a Jordan algebra
It is remarkable that Euclidean Jordan algebras turned out to have such a wide range of applications

Summary

Introduction

In this paper we explore the close and interesting relationship of a three-dimensional Euclidean Jordan algebra to the adjacency matrix of a strongly regular graph X. The parameters of a (n, k, a, c) -strongly regular graph are not independent and are related by the equality k (k − a −1) = (n − k −1) c It is well known (see, for instance, [9]) that the eigenvalues of a (n, k, a, c) -strongly regular graph X are k, θ and τ , where θ and τ are given by ( ) θ = a − c + (a − c)2 + 4(k − c) 2,. In order to obtain new feasible conditions for the existence of a strongly regular graph, in Section 3, we define the generalized Krein parameters of a strongly regular graph. Since the generalized Krein parameters are nonnegative we establish new admissibility conditions, for the parameters of a strongly regular graph that give different information from that given by the Krein conditions 6) - 7)

Euclidean Jordan Algebras and Strongly Regular Graphs

A Generalization of the Krein Parameters

Relations between the Krein Parameters and the Generalized Krein Parameters

Some Conclusions