1 Euclidean Jordan algebras and inequalities on the parameters and on the spectra of a strongly regular graph

By Luis Vieira.

Let τ be a strongly (n, p; a, c) regular graph, such that 0 < c < p < n − 1, A his matrix of adjacency and let V_{n} be the Euclidean real space spanned by the powers of A^{j}, j ∈ ℕ_{0} where the scalar product •|• is defined by x|y = trace(x · y). In this work ones proves that V_{n} is an Euclidean Jordan algebra of rank 3 when one introduces in V_{n} the usual product of matrices. Working inside the Euclidean Jordan algebra V_{n} with the the only complete system of orthogonal idempotents associated to A one defines the generalized krein parameters of the strongly (n, p; a, c) regular graph τ. Finally one presents necessary conditions over the parameters and the spectra of the τ strongly (n, p; a, c) regular graph.