Sharp $A_\alpha$-Spectral Conditions for Odd $[1,b]$-Factors When $\alpha>1/2$

Abstract

We solve, for all sufficiently large even orders, the problem proposed by Chen et al. on sharp $A_\alpha$-spectral conditions for the existence of odd $[1,b]$-factors when $\alpha>1/2$. Chen et al. showed that every connected graph of even order $n$ with no odd $[1,b]$-factor has $A_\alpha$-spectral radius at most $\max_{1\le s\le k}\rho_\alpha(G_s)$, where $G_s=K_s\nabla\left(K_{n-(b+1)s-1}\cup(bs+1)K_1\right)$ and $k=\lfloor(n-2)/(b+1)\rfloor$. Thus the problem reduces to finding the graph with the largest $A_\alpha$-spectral radius among these obstruction graphs. We prove that, for every $\alpha\in(1/2,1)$, $\max_{1\le s\le k}\rho_\alpha(G_s)=\max\{\rho_\alpha(G_1),\rho_\alpha(G_k)\}$. Moreover, for each fixed odd $b\ge 3$ and every even $n\ge N_b=(b+1)\max\{2b+3,14\}+2$, there exists a unique $\alpha=\alpha_\ast(n,b)\in(1/2,1)$ at which $\rho_\alpha(G_1)=\rho_\alpha(G_k)$. Consequently, $G_1$ is the unique extremal graph for $1/2<\alpha<\alpha_\ast(n,b)$, both $G_1$ and $G_k$ are extremal at $\alpha=\alpha_\ast(n,b)$, and $G_k$ is the unique extremal graph for $\alpha_\ast(n,b)<\alpha<1$. This gives the exact $A_\alpha$-spectral threshold, together with the sharp exceptional graphs, for odd $[1,b]$-factors when $\alpha>1/2$ and $n\ge N_b$.