Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21287120220601Two upper bounds on the A_α-spectral radius of a connected graph53571417810.22049/cco.2021.27061.1187ENShariefuddinPirzadaDepartment of Mathematics, HazratbalJournal Article20201225If $A(G)$ 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 $0leq alpha leq 1$. 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
$$ lambda_{alpha}leq alpha~Delta +(1-alpha)sqrt{2mleft(1-frac{1}{omega}right)}, $$<br />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*}<br /> lambda_{alpha}leq frac{1}{2}maxlimits_{1leq ileq n} left[alpha d_{i}+sqrt{ alpha^{2}d_{i}^{2}+4m_{i}(1-alpha)[alpha+(1-alpha)m_{j}] }right],<br />end{equation*}<br />where $d_{i}$ and $m_{i}$ are respectively the degree and the average 2-degree of the vertex $v_{i}$.http://comb-opt.azaruniv.ac.ir/article_14178_9ff8635c0123896b88601cc9982321d9.pdf