Edge-Disjoint Spanning Trees, Eigenvalues, and Size of Graphs

Abstract

The spanning tree packing number of a graph $G$, written by $\tau \left(G\right)$, is defined as the maximum number of edge-disjoint spanning trees. The study of $\tau \left(G\right)$ is a classic problem in graph theory, and this parameter has important application in analyzing network robustness, routing optimization, and fault tolerance. Paul Seymour motivated the academia to establish the relationship between $\tau \left(G\right)$ and eigenvalues of a connected graph. Let $G_{n,\delta }$ be the class of $n$-vertex graphs with minimum degree $\delta$ and let $G\left(n,m,k\right)$ denote the set of graphs of order $n$ and size $m$ admitting exactly $k$ edge-disjoint spanning trees. Fan, Gu, and Lin [J. Graph Theory, 104 (2023) 697-711] gave a sufficient size condition, together with its spectral analogue, to ensure that a graph $G\in G_{n,\delta }$ satisfies $\tau \left(G\right)⩾k$ under the condition $\delta ⩾2k$. It is interesting to see that these problems are still open under the condition $k⩽\delta ⩽2k-1$. In this paper, we fill this gap. Then we address an open problem posed by Fan, Gu, and Lin [J. Graph Theory, 104 (2023) 697-711]: Determine the graphs among $G\left(n,k\left(n-1\right),k\right)$ having maximum spectral radius. Furthermore, they conjectured that $K_k\nabla \left(K_k\cup \left(n-2k\right)K_1\right)$ is the unique graph among $G\left(n,k\left(n-1\right),k\right)$ attaining maximum spectral radius for $n⩾4$ and $k⩾2$. We show that, for $n⩾\frac{k^2+k+2}{2}$ with $k⩾5,K_k\nabla \left(K_{1,\frac{k\left(k-1\right)}{2}}\cup \left(n-\frac{k^2+k+2}{2}\right)K_1\right)$ is the unique graph among $G\left(n,k\left(n-1\right),k\right)$ having maximum spectral radius. This resolves their open problem, and consequently disproves their conjecture.