A spanning tree T of a connected graph G is a subgraph of G that is a tree covering all vertices of G. The leaf distance of T is defined as the minimum of distances between any two leaves of T. A fractional matching of a graph G is a function h assigning every edge a real number in [0, 1] so that \(\sum \limits _{e\in E_G(v)}{h(e)}\le 1\) for any \(v\in V(G)\) , where \(E_G(v)\) denotes the set of edges incident with v in G. A fractional matching of G is called a fractional perfect matching if \(\sum \limits _{e\in E_G(v)}{h(e)}=1\) for any \(v\in V(G)\) . A graph G with at least \(2k+2\) vertices is said to be fractional k-extendable if every k-matching M in G is included in a fractional perfect matching h of G such that \(h(e)=1\) for any \(e\in M\) . This paper considers a lower bound on the spectral radius of G to guarantee that G has a spanning tree with leaf distance at least d. At the same time, we obtain a lower bound on the spectral radius of G to ensure that G is fractional k-extendable.
