@article {
author = {Pirzada, Shariefuddin},
title = {Two upper bounds on the A_α-spectral radius of a connected graph},
journal = {Communications in Combinatorics and Optimization},
volume = {7},
number = {1},
pages = {53-57},
year = {2022},
publisher = {Azarbaijan Shahid Madani University},
issn = {2538-2128},
eissn = {2538-2136},
doi = {10.22049/cco.2021.27061.1187},
abstract = {If $A(G)$ and $D(G)$ are respectively the adjacency matrix and the diagonal matrix of vertex degrees of a connected graph $G$, the generalized adjacency matrix $A_{\alpha}(G)$ is defined as $A_{\alpha}(G)=\alpha ~D(G)+(1-\alpha)~A(G)$, where $0\leq \alpha \leq 1$. The $A_{\alpha}$ (or generalized) spectral radius $\lambda(A_{\alpha}(G))$ (or simply $\lambda_{\alpha}$) is the largest eigenvalue of $A_{\alpha}(G)$. In this paper, we show that $$ \lambda_{\alpha}\leq \alpha~\Delta +(1-\alpha)\sqrt{2m\left(1-\frac{1}{\omega}\right)}, $$where $m$, $\Delta$ and $\omega=\omega(G)$ are respectively the size, the largest degree and the clique number of $G$. Further, if $G$ has order $n$, then we show that \begin{equation*} \lambda_{\alpha}\leq \frac{1}{2}\max\limits_{1\leq i\leq n} \left[\alpha d_{i}+\sqrt{ \alpha^{2}d_{i}^{2}+4m_{i}(1-\alpha)[\alpha+(1-\alpha)m_{j}] }\right],\end{equation*}where $d_{i}$ and $m_{i}$ are respectively the degree and the average 2-degree of the vertex $v_{i}$.},
keywords = {Adjacency matrix,generalized adjacency matrix,spectral radius,clique number},
url = {http://comb-opt.azaruniv.ac.ir/article_14178.html},
eprint = {http://comb-opt.azaruniv.ac.ir/article_14178_9ff8635c0123896b88601cc9982321d9.pdf}
}