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2014 | OriginalPaper | Chapter

38. Inequalities on the Parameters of a Strongly Regular Graph

Authors : Vasco Moço Mano, Enide Andrade Martins, Luís Almeida Vieira

Published in: Modeling, Dynamics, Optimization and Bioeconomics I

Publisher: Springer International Publishing

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Abstract

In this paper we establish inequalities over the parameters and over the spectra of strongly regular graphs in the environment of Euclidean Jordan algebras. We consider a strongly regular graph, G, whose adjacency matrix A has three distinct eigenvalues, and the Euclidean Jordan algebra of real symmetric matrices of order n, \(\text{Sym}(n, \mathbb{R})\) with the vector product and the inner product being the Jordan product and the usual trace of matrices, respectively. We associate a three dimensional real Euclidean Jordan subalgebra \(\mathcal{A}\) of \(\text{Sym}(n, \mathbb{R})\) to A, spanned by the identity matrix and the natural powers of A. Next, we compute the unique complete system of orthogonal idempotents \(\mathcal{B}\) associated to A and we consider particular convergent Hadamard series constructed from the idempotents of \(\mathcal{B}\). Finally, by the analysis of the spectra of the sums of these Hadamard series we establish new conditions for the existence of a strongly regular graph.

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Metadata
Title
Inequalities on the Parameters of a Strongly Regular Graph
Authors
Vasco Moço Mano
Enide Andrade Martins
Luís Almeida Vieira
Copyright Year
2014
DOI
https://doi.org/10.1007/978-3-319-04849-9_38