Spanning subgraphs and spectral radius in graphs

A spanning tree $T$ of a connected graph $G$ is a subgraph of $G$ that is a tree covers 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 $\sum _{e\in E_G(v)}h(e)\le 1$ 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 $\sum _{e\in E_G(v)}h(e)=1$ 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.