On the dominating induced matching problem: Spectral results and sharp bounds

A matching M is a dominating induced matching of a graph if every edge is either in M or has a common end-vertex with exactly one edge in M. The extremal graphs on the number of edges with dominating induced matchings are characterized by its Laplacian spectrum and its principal Laplacian eigenvector. Adjacency, Laplacian and signless Laplacian spectral bounds on the cardinality of dominating induced matchings are obtained for arbitrary graphs. Moreover, it is shown that some of these bounds are sharp and examples of graphs attaining the corresponding bounds are given.