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

The binding number, denoted by $bind(G)$, of a graph $G$ is defined as the minimum value of $\frac{\mathrm{\mid }{N}_G(X)\mathrm{\mid }}{\mathrm{\mid }X\mathrm{\mid }}$ taken over any non-empty subset $X$ of $V(G)$ with $N_G(X)\mathrm{\ne }V(G)$. A graph $G$ is said to be $r$-binding if $bind(G)\ge r$. 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.