Spectral results on regular graphs with (k, τ)-regular sets

Abstract

A set of vertices S icluded in V (G) is (k, τ)-regular if it induces a k-regular subgraph of G such that | NG (v) ∩ S | = τ if v is not in S. Note that a connected graph with more than one edge has a perfect matching if and only if its line graph has a (0, 2)-regular set. In this paper, some spectral results on the adjacency matrix of graphs with (k, τ)-regular sets are presented. Relations between the combinatorial structure of a p-regular graph with a (k, τ)-regular set and the eigenspace corresponding to each eigenvalue λ not in { p, k - τ } are deduced. Finally, additional results on the effects of Seidel switching (with respect to a bipartition induced by S) of regular graphs are also introduced. © 2006 Elsevier B.V. All rights reserved.