Two upper bounds on the A_α-spectral radius of a connected graph

Document Type : Original paper

Author

Department of Mathematics, Hazratbal

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 $v_{i}$.

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References

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