Binomial Hadamard Series and Inequalities over the Spectra of a Strongly Regular Graph

Luís Vieira

doi:10.4236/am.2018.99071

Abstract

Let G be a primitive strongly regular graph of order n and A is adjacency matrix. In this paper we first associate to A a real 3-dimensional Euclidean Jordan algebra with rank three spanned by In and the natural powers of A that is a subalgebra of the Euclidean Jordan algebra of symmetric matrix of order n. Next we consider a basis that is a Jordan frame of . Finally, by an algebraic asymptotic analysis of the second spectral decomposition of some Hadamard series associated to A we establish some inequalities over the spectra and over the parameters of a strongly regular graph.

Highlights

Good surveys on Euclidean Jordan algebras can be found in books such as A Taste of Jordan Algebras of Kevin McCrimmon [1], Analysis on Symmetric Cones of Faraut and Korányi [2] and in the Koecher’s Minnesota Notes on Jordan Algebras and Their Applications [3]
By an algebraic asymptotic analysis of the second spectral decomposition of some Hadamard series associated to A we establish some inequalities over the spectra and over the parameters of a strongly regular graph
We make an introduction to Euclidean Jordan algebras and present the definitions and the more relevant results needed for this paper without presenting the proofs of the results presented in this paper, since they are very well deduced in the monograph Analysis on symmetric cones of Faraut and Korányi, see [2]

Summary

Introduction

Good surveys on Euclidean Jordan algebras can be found in books such as A Taste of Jordan Algebras of Kevin McCrimmon [1], Analysis on Symmetric Cones of Faraut and Korányi [2] and in the Koecher’s Minnesota Notes on Jordan Algebras and Their Applications [3]. Euclidean Jordan algebras become a good tool for the analysis of primal dual interior point methods [4] [5] [6] and [7]. They have a lot of applications on other branches of mathematics namely on the formalism of quantum mechanics [8], on combinatorics [9]-[15], and on statistics [16]. In this paper we analyse the spectra of Hadamard series associated to the adjacency matrix of a strongly regular graph to deduce asymptotic inequalities on the spectra and on the parameters of a strongly regular graph in the environment of Euclidean Jordan algebras.

A Short Introduction to Euclidean Jordan Algebras

Some Notions on Strongly Regular Graphs