  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.
