Adjacency spectral radius and H-factors in 1-binding graphs (2506.20273v1)

Abstract: Let $G$ be a graph, and let $H:V(G)\longrightarrow{{1},{0,2}}$ be a set-valued function. Hence, $H(v)$ equals ${1}$ or ${0,2}$ for any $v\in V(G)$. We let $$ H^{-1}(1)={v: v\in V(G) \ \mbox{and} \ H(v)=1}. $$ 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 $n\geq11$ satisfies $\rho(G)\geq\rho(K_1\vee(K_{n-4}\cup K_2\cup K_1))$, then $G$ has an $H$-factor for each $H:V(G)\longrightarrow{{1},{0,2}}$ with $H^{-1}(1)$ even, unless $G=K_1\vee(K_{n-4}\cup K_2\cup K_1)$.