Spectral bounds for the k-regular induced subgraph problem

Abstract

Many optimization problems on graphs are reduced to the determination of a subset of vertices of maximum cardinality which induces a $k$-regular subgraph. For example, a maximum independent set, a maximum induced matching and a maximum clique is a maximum cardinality $0$-regular, $1$-regular and $(\omega(G)-1)$-regular induced subgraph, respectively, were $\omega(G)$ denotes the clique number of the graph $G$. The determination of the order of a $k$-regular induced subgraph of highest order is in general an NP-hard problem. This paper is devoted to the study of spectral upper bounds on the order of these subgraphs which are determined in polynomial time and in many cases are good approximations of the respective optimal solutions. The introduced upper bounds are deduced based on adjacency, Laplacian and signless Laplacian spectra. Some analytical comparisons between them are presented. Finally, all of the studied upper bounds are tested and compared through several computational experiments.