Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6423397 | Discrete Mathematics | 2014 | 6 Pages |
For a graph G, let L(G) and Q(G) be the Laplacian and signless Laplacian matrices of G, respectively, and Ï(G) be the number of spanning trees of G. We prove that if G has an odd number of vertices and Ï(G) is not divisible by 4, then (i) L(G) has no even integer eigenvalue, (ii) Q(G) has no integer eigenvalue λâ¡2(mod4), and (iii) Q(G) has at most one eigenvalue λâ¡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 Ï(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 Ï(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.