Euclidean Jordan algebras and strongly regular graphs

Abstract

We analyze the spectra of strongly regular graphs in the environment of Euclidean Jordan algebras. In particular we obtain the spectra of the strongly regular graphs constructed in the Euclidean Jordan algebra studied in Cardoso and Vieira (J Math Sci 120:881-894, 2004) recurring to homogeneous linear difference equations of second order with constant coefficients. Next, we associate a three dimensional Euclidean Jordan algebra V to the adjacency matrix of a strongly regular graph τ with three distinct eigenvalues and we define the generalized Krein parameters of τ. Finally, we establish necessary conditions for the existence of a strongly regular graph. © Springer-Verlag Berlin Heidelberg 2011.