An Aα-spectral radius for a spanning tree with constrained leaf distance in a graph

Abstract

Let $\alpha\in[0,1)$ be a real number, and let $G$ be a connected graph of order $n$ with $n\geq\lambda(\alpha)$, where $\lambda(\alpha)=9$ for $0\leq\alpha\leq\frac{2}{3}$ and $\lambda(\alpha)=\frac{4}{1-\alpha}$ for $\frac{2}{3 <\alpha<1$. A spanning tree $T$ of $G$ is a subgraph of $G$
that is a tree covers all vertices of $G$. The leaf distance of a tree is the minimum of distances between any two leaves of a tree. Let
$A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$, where $A(G)$ is the adjacency matrix of $G$ and $D(G)$ is the diagonal matrix of vertex degrees of $G$. The largest eigenvalues of $A_{\alpha}(G)$, denoted by $\rho_{\alpha}(G)$, is called $A_{\alpha}$-spectral radius of $G$. In this paper, it is proved that $G$ has a spanning tree with leaf distance at least 4 if $\rho_{\alpha (G)\geq\gamma(n)$, where $\gamma(n)$ is the largest root of
$x^{3}-(\alpha n+n+\alpha-3)x^{2}+(\alpha n^{2}+\alpha^{2}n-\alpha n-n-2\alpha+1)x-\alpha^{2}n^{2}+3\alpha^{2}n-\alpha n+n 4\alpha^{2}+5\alpha-3=0$.