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 ( mod 4 ) , and (iii) Q ( G ) has at most one eigenvalue λ ≡ 0 ( mod 4 ) 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 ) = 2 t s 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.