Spanning trees and even integer eigenvalues of graphs

For a graph G, let L(G) and Q(G) be the Laplacian and signless Laplacian matrices of G, respectively, and τ(G) be the number of spanning trees of G. We prove that if G has an odd number of vertices and τ(G) is not divisible by 4, then (i) L(G) has no even integer eigenvalue, (ii) Q(G) has no integer eigenvalue λ≡2(mod4), and (iii) Q(G) has at most one eigenvalue λ≡0(mod4) and such an eigenvalue is simple. As a consequence, we extend previous results by Gutman and Sciriha and by Bapat on the nullity of adjacency matrices of the line graphs. We also show that if τ(G)=2ts with s odd, then the multiplicity of any even integer eigenvalue of Q(G) is at most t+1. Among other things, we prove that if L(G) or Q(G) has an even integer eigenvalue of multiplicity at least 2, then τ(G) is divisible by 4. As a very special case of this result, a conjecture by Zhou et al. (2013) on the nullity of adjacency matrices of the line graphs of unicyclic graphs follows.