  For a graph $G$, let $L(G)$ and $Q(G)$ be the Laplacian and signless
Laplacian matrices of $G$, respectively, and $\tau(G)$ be the number of
spanning trees of $G$. We prove that if $G$ has an odd number of vertices and
$\tau(G)$ is not divisible by $4$, then (i) $L(G)$ has no even integer
eigenvalue, (ii) $Q(G)$ has no integer eigenvalue $\lambda\equiv2\pmod4$, and
(iii) $Q(G)$ has at most one eigenvalue $\lambda\equiv0\pmod4$ 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 $\tau(G)=2^ts$ 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 $\tau(G)$ is divisible by $4$. As a very
special case of this result, a conjecture by Zhou et al. [On the nullity of
connected graphs with least eigenvalue at least $-2$, Appl. Anal. Discrete
Math. 7 (2013), 250--261] on the nullity of adjacency matrices of the line
graphs of unicyclic graphs follows.
