Sharp upper bounds on the $A_{\hat{I}}\pm$-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 $\hat{I}\pm \in [0,1]$, Nikiforov (2017) proposed the $A_{\hat{I}}\pm$-matrix of a graph $G$ as $A_{\hat{I}}\pm (G)=\hat{I}\pm D(G)+(1-\hat{I}\pm )A(G)$. The $A_{\hat{I}}\pm$-spectral radius of $G$, denoted by $\ddot{I}{}_{\hat{I}}\pm (G)$, is the largest eigenvalue of $A_{\hat{I}}\pm (G)$, where $\ddot{I}{}_0(G)=\ddot{I}(G)$ is the spectral radius of $A(G)$ and $2\ddot{I}{}_{\frac{1}{2}}(G)=q(G)$ is the spectral radius of $Q(G)$. Sun and Das (2020) proved that for any non-isolated vertex $v$ of degree $d_v$, $\ddot{I}{}^2(G)-\ddot{I}{}^2(G-v)\le 2d_v-1$, 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 $\ddot{I}{}_{\hat{I}}\pm (G)$. As a corollary, for every $k\in [0,d_v+1]$, we have $$\ddot{I}{}^2(G)-\ddot{I}{}^2(G-v)\le 2d_v-1+(k-2)\left(\frac{d_v}{\ddot{I}(G)}-1\right).$$ This inequality coincides with that of Sun and Das when $k=2$, and is strictly sharper than theirs whenever $k\ne 2$ and $d_v\ne \ddot{I}(G)$. We also give a short proof of the inequality $\ddot{I}{}_{\hat{I}}\pm (G)-\ddot{I}{}_{\hat{I}}\pm (G-v)\le \hat{I}\pm +\frac{(1-\hat{I}\pm {)}^2{d}_v}{\ddot{I}{}_{\hat{I}}\pm (G)-\hat{I}\pm d_v}$, which is obtained by Wang and She (2022). Moreover, we obtain a unified generalization of Hong, Shu and Fang's inequality for $\ddot{I}(G)$ and Nikiforov's inequality for $q(G)$ in terms of $\ddot{I}{}_{\hat{I}}\pm (G)$.

12 pages