Main eigenvalues and (κ, τ)-regular sets

A (kappa, tau)-regular set is a subset of the vertices of a graph G, inducing a kappa-regular subgraph such that every vertex not in the subset has tau neighbors in it. A main eigenvalue of the adjacency matrix A of a graph G has an eigenvector not orthogonal to the all-one vector j. For graphs with a (kappa, tau)-regular set a necessary and sufficient condition for an eigenvalue be non-main is deduced and the main eigenvalues are characterized. These results are applied to the construction of infinite families of bidegreed graphs with two main eigenvalues and the same spectral radius (index). Some relations with strongly regular graphs are also obtained. Finally, the determination of (kappa, tau)-regular sets is discussed. (C) 2009 Elsevier Inc. All rights reserved.