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 $n\ge N_b$.