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 $δ\geq6$, where $n\geq3δ$. In this paper, we give a sharp spectral radius condition for $G$ to contain $k$ edge-disjoint 2-connected factors and $\left\lfloor\frac{δ-4k}{2}\right\rfloor$ edge-disjoint spanning trees, where $1\leq k\leq\left\lfloor\fracδ{4}\right\rfloor$ is an integer.