  Partially answering a question of Paul Seymour, we obtain a sufficient
eigenvalue condition for the existence of $k$ edge-disjoint spanning trees in a
regular graph, when $k\in \{2,3\}$. More precisely, we show that if the second
largest eigenvalue of a $d$-regular graph $G$ is less than
$d-\frac{2k-1}{d+1}$, then $G$ contains at least $k$ edge-disjoint spanning
trees, when $k\in \{2,3\}$. We construct examples of graphs that show our
bounds are essentially best possible. We conjecture that the above statement is
true for any $k<d/2$.
