A note on the Q-integrability of complete bipartite graphs with self-loops

A graph [Formula: see text] is [Formula: see text]-integral if its signless Laplacian has only integral eigenvalues. Suppose [Formula: see text] be the graph after adding a self-loop to every vertex of a subset [Formula: see text] of [Formula: see text]. In this paper, we prove that for a nonempty proper subset [Formula: see text] of [Formula: see text], [Formula: see text] is [Formula: see text]-integral if and only if [Formula: see text] and self-loops are added in exactly two vertices of each partite set, or [Formula: see text] and self-loops are added to all the vertices of one partite set. Besides, we prove that if [Formula: see text] is a nonempty proper subset of [Formula: see text], then [Formula: see text] is [Formula: see text]-integral if and only if [Formula: see text] and self-loops are added to all the pendant vertices. Using these results, we also answer when the matrix [Formula: see text] have only integral spectrum, where [Formula: see text] is any [Formula: see text]-diagonal matrix and [Formula: see text] is permutation similar to either [Formula: see text] or [Formula: see text].