Spectral radius for the existence of $H_b$-factors in binding graphs

Abstract

The binding number, denoted by $\mbox{bind}(G)$, of a graph $G$ is defined as the minimum value of $\frac{|N_G(X)|}{|X|}$ taken over any non-empty subset $X$ of $V(G)$ with $N_G(X)\ne V(G)$. A graph $G$ is said to be $r$-binding if $\mbox{bind}(G)\geq r$. The adjacency matrix of a graph $G$ is denoted by $A(G)$. The largest eigenvalue of $A(G)$ is called the spectral radius of $G$. An $H_b$-factor of a graph $G$ is defined as a spanning subgraph $F$ of $G$ such that for any $v\in V(G)$, $d_F(v)$ belongs to the set $\{1,3,5,\dots ,b-1,b\}$, where $b$ is an even integer with $b\ge 2$. This note establishes a sufficient condition to guarantee that a connected $\frac{1}{b-1}$-binding graph $G$ of even order contains an $H_b$-factor based on the spectral radius.

8 pages