Euclidean Jordan Algebras and Generalized Krein parameters of a strongly regular graph

TL;DR

This paper demonstrates that the space spanned by powers of the adjacency matrix of a strongly regular graph forms a Euclidean Jordan algebra of rank 3, leading to new generalized Krein parameters and spectral conditions.

Contribution

It introduces a novel algebraic framework using Euclidean Jordan algebras to analyze strongly regular graphs and defines generalized Krein parameters with associated spectral conditions.

Findings

The space of adjacency matrix powers forms a Euclidean Jordan algebra of rank 3.

Defines generalized Krein parameters based on Jordan algebra properties.

Provides necessary conditions on graph parameters and spectra.

Abstract

Let $\tau$ be a strongly $(n,p;a,c)$ regular graph,such that $0<c<p<n-1,$ $A$ his matrix of adjacency and let ${\cal V}_{n}$ be the Euclidean space spanned by the powers of $A$ over the reals where the scallar product $\bullet|\bullet$ is defined by $x|y={trace}(x \cdot y).$ In this work ones proves that ${\cal V}_{n}$ is an Euclidean Jordan algebra of rank 3 when one introduces in ${\cal V}_{n}$ the usual product of matrices. In this Euclidean Jordan algebra one defines the modulus of a matrix, and afterwards one defines $|A|^x \forall x\in \mathbb{R}.$ Working inside the Euclidean Jordan algebra ${\cal V}_{n}$ and making use of the properties of $|A|^x$ one defines the generalized krein parameters of the strongly $(n,p;a,c)$ regular graph $\tau$ and finally one presents necessary conditions over the parameters and the spectra of the $\tau$ strongly $(n,p;a,c)$ regular graph.