Spectral Radius and Edge-Disjoint Spanning Trees of Graphs With Prescribed Edge Connectivity

Abstract

Let ${\kappa }^{\prime }\left(G\right)$ and $\tau \left(G\right)$ denote the edge connectivity and the maximum number of edge-disjoint spanning trees contained in a graph $G$, respectively. Fan, Gu, and Lin studied a tight spectral condition of a connected graph to guarantee $\tau \left(G\right)\ge k$. The famous theorem of Tutte and Nash-Williams implies that ${\kappa }^{\prime }\left(G\right)$ and $\tau \left(G\right)$ are closely related with $\tau \left(G\right)\ge \lfloor \frac{{\kappa }^{\prime }\left(G\right)}{2}\rfloor$, so it is interesting to explore conditions on a graph $G$ with ${\kappa }^{\prime }\left(G\right)\le 2k-1$ to warrant $\tau \left(G\right)\ge k$. Since graphs with $k$ edge-disjoint spanning trees are $k$-edge-connected, the converse does not hold. We initially provide a tight spectral radius condition to guarantee $\tau \left(G\right)\ge k$ in $k$-edge-connected graphs with a unique extremal graph characterization. Moreover, an extensive spectral condition on $\tau \left(G\right)\ge k$ in $k$-edge-connected graphs with fixed minimum degree $\delta \ge 4k$ is determined. For $\tau \left(G\right)\ge 2$, we further study analogous results on 3-edge-connected graphs to ensure $\tau \left(G\right)\ge 2$, which implies comprehensive spectral arguments on $\tau \left(G\right)\ge 2$ for ${\kappa }^{\prime }\left(G\right)<4$.