For $0\le \alpha <1$, let $A_{\alpha }\left(G\right)=\alpha D\left(G\right)+(1-\alpha )A\left(G\right)$ be the $A_{\alpha }$-matrix of $G$, where $D\left(G\right)$ is the diagonal degree matrix of $G$ and $A\left(G\right)$ is the adjacency matrix of $G$ Let ${\lambda }_{\alpha }\left(G\right)$ denote the $A_{\alpha }$-spectral radius of a graph $G$. Let $G$ be a $k$-connected nonregular graph with $n$ vertices, $m$ edges, maximum degree $\Delta$ and minimum degree $\delta$. In this paper, we show that $\Delta -{\lambda }_{\alpha }\left(G\right)>\frac{(1-\alpha )(n\Delta -2m)k^2}{(n\Delta -2m)\left[{(n-1)}^2-(\Delta -k+1)(n-k-1)\right]+(1-\alpha )n{k}^2},$which extends the result on the spectral radius of Xue and Liu (0000) and improves the result on the signless Laplacian spectral radius of Shiu et al. (2017). Furthermore, we also prove that $\Delta -{\lambda }_{\alpha }\left(G\right)>\frac{(1-\alpha )(n\Delta -2m)k^2}{(n\Delta -2m)\left[(n-\Delta +2k-2)(n-\delta -1)+k^2\right]+(1-\alpha )n{k}^2},$which extends the result on the signless Laplacian spectral radius of Ning et al. (2018).

Let $\Gamma$ be a $k$-strong nonregular digraph of order $n$, size $m$, and maximum outdegree ${\Delta }^{+}$. For $0\le \alpha <1$, let $A_{\alpha }\left(\Gamma \right)=\alpha D\left(\Gamma \right)+(1-\alpha )A\left(\Gamma \right)$ be the $A_{\alpha }$-matrix of $\Gamma ,$ where $D\left(\Gamma \right)$ is the diagonal matrix of its vertex outdegrees and $A\left(\Gamma \right)$ is the adjacency matrix of $\Gamma .$ Denote by ${\lambda }_{\alpha }\left(\Gamma \right)$ the $A_{\alpha }$-spectral radius of a digraph $\Gamma$. We prove that ${\Delta }^{+}-{\lambda }_{\alpha }\left(\Gamma \right)>\frac{(1-\alpha )(n{\Delta }^{+}-m)k^2}{2(n{\Delta }^{+}-m)\left[{(n-1)}^2-({\Delta }^{+}-k+1)(n-k-1)\right]+(1-\alpha )n{k}^2},$which improves the result of Xi and Wang (2020). In the end of the paper, related problem is mentioned.

为了$0\le \alpha <\mathrm{1个}$， 让${\mathrm{一种}}_{\alpha }\left(G\right)=\alpha 丁\left(G\right)+(\mathrm{1个}-\alpha )\mathrm{一种}\left(G\right)$成为${\mathrm{一种}}_{\alpha }$-矩阵$G$， 在哪里$丁\left(G\right)$是的对角线度矩阵$G$和$\mathrm{一种}\left(G\right)$是邻接矩阵$G$让${\lambda }_{\alpha }\left(G\right)$表示${\mathrm{一种}}_{\alpha }$-图的谱半径$G$. 让$G$是一个$k$-连接的非常规图$n$顶点，$米$边缘，最大程度$△$和最低学历$\delta$. 在本文中，我们表明$△-{\lambda }_{\alpha }\left(G\right)>\frac{(\mathrm{1个}-\alpha )(n△-\mathrm{2个}米)k^{\mathrm{2个}}}{(n△-\mathrm{2个}米)\left[{(n-\mathrm{1个})}^{\mathrm{2个}}-(△-k+\mathrm{1个})(n-k-\mathrm{1个})\right]+(\mathrm{1个}-\alpha )n{k}^{\mathrm{2个}}},$它扩展了 Xue 和 Liu (0000) 谱半径的结果，改进了 Shiu 等人的无符号拉普拉斯谱半径的结果。(2017)。此外，我们还证明$△-{\lambda }_{\alpha }\left(G\right)>\frac{(\mathrm{1个}-\alpha )(n△-\mathrm{2个}米)k^{\mathrm{2个}}}{(n△-\mathrm{2个}米)\left[(n-△+\mathrm{2个}k-\mathrm{2个})(n-\delta -\mathrm{1个})+k^{\mathrm{2个}}\right]+(\mathrm{1个}-\alpha )n{k}^{\mathrm{2个}}},$这扩展了 Ning 等人的无符号拉普拉斯谱半径的结果。(2018)。

让$\Gamma$是一个$k$-强非常规有序有向图$n$， 尺寸$米$, 和最大出度${△}^{+}$. 为了$0\le \alpha <\mathrm{1个}$， 让${\mathrm{一种}}_{\alpha }\left(\Gamma \right)=\alpha 丁\left(\Gamma \right)+(\mathrm{1个}-\alpha )\mathrm{一种}\left(\Gamma \right)$成为${\mathrm{一种}}_{\alpha }$-矩阵$\Gamma ,$在哪里$丁\left(\Gamma \right)$是顶点出度的对角矩阵，$\mathrm{一种}\left(\Gamma \right)$是邻接矩阵$\Gamma .$表示为${\lambda }_{\alpha }\left(\Gamma \right)$这${\mathrm{一种}}_{\alpha }$-有向图的谱半径$\Gamma$. 我们证明${△}^{+}-{\lambda }_{\alpha }\left(\Gamma \right)>\frac{(\mathrm{1个}-\alpha )(n{△}^{+}-米)k^{\mathrm{2个}}}{\mathrm{2个}(n{△}^{+}-米)\left[{(n-\mathrm{1个})}^{\mathrm{2个}}-({△}^{+}-k+\mathrm{1个})(n-k-\mathrm{1个})\right]+(\mathrm{1个}-\alpha )n{k}^{\mathrm{2个}}},$这改进了 Xi 和 Wang (2020) 的结果。在论文的最后，提到了相关问题。