Adjacency spectral radius and H-factors in 1-binding graphs

Let $G$ be a graph, and let $H:V(G)⟶\{\{1\},\{0,2\}\}$ be a set-valued function. Hence, $H(v)$ equals $\{1\}$ or $\{0,2\}$ for any $v\in V(G)$. We let $$H^{-1}(1)=\{v:v\in V(G)andH(v)=1\}\mathrm{.}$$ An $H$-factor of $G$ is a spanning subgraph $F$ of $G$ such that $d_F(v)\in H(v)$ for each $v\in V(G)$. Lu and Kano showed a characterization for the existence of an $H$-factor in a graph [Characterization of 1-tough graphs using factors, Discrete Math. 343 (2020) 111901]. Let $A(G)$ and $\rho (G)$ denote the adjacency matrix and the adjacency spectral radius of $G$, respectively. By using Lu and Kano’s result, we pose a sufficient condition with respect to the adjacency spectral radius to guarantee the existence of an $H$-factor in a 1-binding graph. In this paper, we prove that if a connected 1-binding graph $G$ of order $n\ge 11$ satisfies $\rho (G)\ge \rho (K_1\vee (K_{n-4}\cup K_2\cup K_1))$, then $G$ has an $H$-factor for each $H:V(G)⟶\{\{1\},\{0,2\}\}$ with $H^{-1}(1)$ even, unless $G=K_1\vee (K_{n-4}\cup K_2\cup K_1)$.