Sharp upper bounds on the $A_\alpha$-spectral radius of graphs

Abstract

Let $G$ be a simple graph with degree diagonal matrix $D(G)$ and adjacency matrix $A(G)$. The signless Laplacian matrix of $G$ is defined as $Q(G)=D(G)+A(G)$. For a real number $\alpha \in [0, 1]$, Nikiforov (2017) proposed the $A_\alpha$-matrix of a graph $G$ as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$. The $A_\alpha$-spectral radius of $G$, denoted by $\rho_\alpha(G)$, is the largest eigenvalue of $A_\alpha(G)$, where $\rho_0(G)=\rho(G)$ is the spectral radius of $A(G)$ and $2\rho_{\frac{1}{2}}(G)=q(G)$ is the spectral radius of $Q(G)$. Sun and Das (2020) proved that for any non-isolated vertex $v$ of degree $d_v$, $\rho^2(G)-\rho^2(G-v) \leq 2 d_v-1$, which confirmed the conjecture originally posed by Guo, Wang, and Li (2019). Recently, Liu and Ning (2026) provided a short and self-contained proof of this inequality. In this paper, we establish the corresponding result for $\rho_\alpha(G)$. As a corollary, for every $k\in [0,d_v+1]$, we have $$ \rho^2(G)- \rho^2(G-v) \leq 2d_v-1 +(k-2)\left(\frac{d_v}{\rho(G)}-1\right). $$ This inequality coincides with that of Sun and Das when $k=2$, and is strictly sharper than theirs whenever $k\neq 2$ and $d_v\neq \rho(G)$. We also give a short proof of the inequality $\rho_{\alpha}(G)-\rho_{\alpha}(G-v)\leq \alpha +\frac{(1-\alpha)^2d_v}{\rho_{\alpha}(G)-\alpha d_v}$, which is obtained by Wang and She (2022). Moreover, we obtain a unified generalization of Hong, Shu and Fang's inequality for $\rho(G)$ and Nikiforov's inequality for $q(G)$ in terms of $\rho_\alpha(G)$.