A note on graphs whose signless Laplacian has three distinct eigenvalues

Abstract

We investigate graphs whose signless Laplacian matrix has three distinct eigenvalues. We show that the largest signless Laplacian eigenvalue of a connected graph G with three distinct signless Laplacian eigenvalues is noninteger if and only if G = K n − e for n ≥ 4, where K n − e is the n vertex complete graph with an edge removed. Moreover, examples of such graphs are given in this article.