Spectral results on regular graphs with \((k,\tau)\)-regular sets (Q878619)

A set of vertices \(S\subseteq V(G)\) is \((k,\tau)\)-regular if it induces a regular subgraph of \(G\) such that \(| N_G(v)\cap S| =\tau\) whenever \(v\notin S\). In this paper, some spectral results on the adjacency matrix of graphs with \((k,\tau)\)-regular sets are presented. Relations between the combinatorial structure of a \(p\)-regular graph with a \((k,\tau)\)-regular set and the eigenspace corresponding to each eigenvalue \(\lambda\notin\{p,k{-}\tau\}\) are deduced. Finally, results on the effect of Seidel switching (with respect to a bipartition induced by \(S\)) of regular graphs are also introduced.