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

TL;DR

This paper establishes eigenvalue conditions under which a graph contains multiple edge-disjoint spanning trees and an additional forest with specific edge constraints, extending previous spectral graph theory results.

Contribution

It introduces new eigenvalue conditions that guarantee the existence of multiple edge-disjoint spanning trees and a forest with prescribed edge properties, including component size constraints.

Findings

Eigenvalue conditions imply $k$ edge-disjoint spanning trees.

Existence of a forest with more than rac{ ext{delta}-1}{ ext{delta}}(|V(G)|-1) edges.

If the forest is not a spanning tree, it has a component with at least delta edges.

Abstract

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.