Spectral results for the dominating induced matching problem

Abstract

A matching M is a dominating induced matching of a graph, if every edge of the graph is either in $M$ or has a common end-vertex with exactly one edge in $M$. The concept of complete dominating induced matching is introduced as graphs where the vertex set can be partitioned into two subsets, one of them inducing an $1$-regular graph and the other defining an independent set and such that all the remaining edges connect each vertex of one set to each vertex of the other. The principal eigenvectors of the adjacency, Laplacian and signless Laplacian matrices of graphs with complete dominating induced matchings are characterized and, therefore, the polynomial time recognition of graphs with complete dominating induced matchings is stated. The adjacency, Laplacian and signless Laplacian spectrum of graphs with complete dominating induced matchings are characterized. Finally, several upper and lower bounds on the cardinality of a dominating induced matching obtained from the eigenvalues of the adjacency, Laplacian and signless Laplacian matrices are deduced and examples for which some of these bounds are tight are presented.