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

Abstract

A graph G is Q-integral if its signless Laplacian has only integral eigenvalues. Suppose GH be the graph after adding a self-loop to every vertex of a subset H of V (G). In this paper, we prove that for a nonempty proper subset H of V (Kp,p), Kp,pH is Q-integral if and only if p = 4 and self-loops are added in exactly two vertices of each partite set, or p = 2 and self-loops are added to all the vertices of one partite set. Besides, we prove that if H is a nonempty proper subset of V (K1,p), then K1,pH is Q-integral if and only if p = 4 and self-loops are added to all the pendant vertices. Using these results, we also answer when the matrix M + 3D′ have only integral spectrum, where D′ is any (0, 1)-diagonal matrix and M is permutation similar to either (Op J J Op ) or (pJ Jip). © 2024 World Scientific Publishing Company.