There are exactly 172 connected Q-integral graphs up to 10 vertices

Abstract

A graph is called $Q$-integral if its signless Laplacian spectrum consists entirely of integers. We establish that there are exactly 172 connected $Q$-integral graphs up to 10 vertices. Pictures or adjacency matrices of those graphs, their $Q$-spectra, some data and comments are given. In addition, we present the connected graphs of the smallest order (which are neither regular nor complete bipartite) being integral in the sense of each of the following three spectra: usual one (related to the adjacency matrix), Laplacian and signless Laplacian.