Spanning tree packing, edge-connectivity and eigenvalues of graphs with given girth

Abstract

Let $\tau (G)$ and ${\kappa }^{\prime }(G)$ denote the edge-connectivity and the spanning tree packing number of a graph $G$, respectively. Proving a conjecture initiated by Cioaba and Wong, Liu et al. in 2014 showed that for any simple graph $G$ with minimum degree $\delta \ge 2k\ge 4$, if the second largest adjacency eigenvalue of $G$ satisfies ${\lambda }_2(G)<\delta -\frac{2k-1}{\delta +1}$, then $\tau (G)\ge k$. Similar results involving the Laplacian eigenvalues and the signless Laplacian eigenvalues of $G$ are also obtained. In this paper, we find a function $f(\delta ,k,g)$ such that for every graph $G$ with minimum degree $\delta \ge 2k\ge 4$ and girth $g\ge 3$, if its second largest adjacency eigenvalue satisfies ${\lambda }_2(G)<f(\delta ,k,g)$, then $\tau (G)\ge k$. As $f(\delta ,k,3)=\delta -\frac{2k-1}{\delta +1}$, this extends the above-mentioned result of Liu et al. Related results involving the girth of the graph, Laplacian eigenvalues and the signless Laplacian eigenvalues to describe $\tau (G)$ and ${\kappa }^{\prime }(G)$ are also obtained.