Aα-spectral radius and path-factor covered graphs

Abstract

Let $\alpha \in [0,1)$, and let $G$ be a connected graph of order $n$ with $n\ge f(\alpha )$, where $f(\alpha )=14$ for $\alpha \in [0,\frac{1}{2}]$, $f(\alpha )=17$ for $\alpha \in (\frac{1}{2},\frac{2}{3}]$, $f(\alpha )=20$ for $\alpha \in (\frac{2}{3},\frac{3}{4}]$ and $f(\alpha )=\frac{5}{1-\alpha }+1$ for $\alpha \in (\frac{3}{4},1)$. A path factor is a spanning subgraph $F$ of $G$ such that every component of $F$ is a path with at least two vertices. Let $k\ge 2$ be an integer. A $P_{\ge k}$-factor means a path-factor with each component being a path of order at least $k$. A graph $G$ is called a $P_{\ge k}$-factor covered graph if $G$ has a $P_{\ge k}$-factor containing $e$ for any $e\in E(G)$. Let $A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G)$, where $D(G)$ denotes the diagonal matrix of vertex degrees of $G$ and $A(G)$ denotes the adjacency matrix of $G$. The largest eigenvalue of $A_{\alpha }(G)$ is called the $A_{\alpha }$-spectral radius of $G$, which is denoted by ${\rho }_{\alpha }(G)$. In this paper, it is proved that $G$ is a $P_{\ge 2}$-factor covered graph if ${\rho }_{\alpha }(G)>\eta (n)$, where $\eta (n)$ is the largest root of $x^3-((\alpha +1)n+\alpha -4)x^2+(\alpha n^2+({\alpha }^2-2\alpha -1)n-2\alpha +1)x-{\alpha }^2{n}^2+(5{\alpha }^2-3\alpha +2)n-10{\alpha }^2+15\alpha -8=0$. Furthermore, we provide a graph to show that the bound on $A_{\alpha }$-spectral radius is optimal.

18 pages