Some Results on \(Q\)-Integral Graphs

Abstract

We consider the problem of determining the \(Q\)-integral graphs, i.e., the graphs with integral signless Laplacian spectrum. First, we determine some infinite series of such graphs having the other two spectra (the usual one and the Laplacian) integral. We also completely determine all \((2, s)\)-semiregular bipartite graphs with integral signless Laplacian spectrum. Finally, we give some results concerning \((3, 4)\) and \((3, 5)\)-semiregular bipartite graphs with the same property.