Eigenvalue conditions implying edge-disjoint spanning trees and a forest with constraints

Let $G$ be a nontrivial graph with minimum degree $\delta$ and $k$ an integer with $k\ge 2$. In the literature, there are eigenvalue conditions that imply $G$ contains $k$ edge-disjoint spanning trees. We give eigenvalue conditions that imply $G$ contains $k$ edge-disjoint spanning trees and another forest $F$ with $|E(F)|>\frac{\delta-1}{\delta}(|V(G)|-1)$, and if $F$ is not a spanning tree, then $F$ has a component with at least $\delta$ edges.