Spectral radius and edge-disjoint connected factors of graphs

Abstract

For a graph $G$, the spectral radius of $G$ is the largest eigenvalue of its adjacency matrix. A connected factor of $G$ is a connected spanning subgraph of $G$. For example, a spanning tree of $G$ is a 1-connected factor of $G$. Let $G$ be a graph of order $n$ with minimum degree $\delta \ge 6$, where $n\ge 3\delta$. In this paper, we give a sharp spectral radius condition for $G$ to contain $k$ edge-disjoint 2-connected factors and $\lfloor \frac{\delta -4k}{2}\rfloor$ edge-disjoint spanning trees, where $1\le k\le \lfloor \frac{\delta }{4}\rfloor$ is an integer.