Edge Connectivity, Packing Spanning Trees, and Eigenvalues of Graphs

TL;DR

This paper investigates the relationship between eigenvalues of graphs, specifically the third largest eigenvalue, and edge connectivity and spanning tree packing numbers in certain classes of simple and multigraphs.

Contribution

It establishes new bounds connecting the third largest eigenvalue with edge connectivity and spanning tree packings in graphs within a specific class.

Findings

Bound on third largest eigenvalue related to edge connectivity

Bound on third largest eigenvalue related to spanning tree packings

Results apply to both simple graphs and multigraphs

Abstract

Let $\mathcal{G}$ be the set of simple graphs (or multigraphs) $G$ such that for each $G \in \mathcal{G}$ there exists at least two non-empty disjoint proper subsets $V_{1},V_{2}\subseteq V(G)$ satisfying $V(G)\setminus(V_{1} \cup V_{2})\neq \phi$ and edge connectivity $\kappa'(G)=e(V_{i},V(G)\backslash V_{i})$ for $1\leq i \leq 2$. A multigraph is a graph with possible multiple edges, but no loops. Let $\tau(G)$ be the maximum number of edge-disjoint spanning trees of a graph $G$. Motivated by a question of Seymour on the relationship between eigenvalues of a graph $G$ and bounds of $\tau(G)$, we mainly give the relationship between the third largest (signless Laplacian) eigenvalue and the bound of $\kappa'(G)$ and $\tau(G)$ of a simple graph or a multigraph $G\in\mathcal{G}$, respectively.