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Two upper bounds on the A_α-spectral radius of a connected graph | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 5، دوره 7، شماره 1، شهریور 2022، صفحه 53-57 اصل مقاله (335.84 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2021.27061.1187 | ||
| نویسنده | ||
| Shariefuddin Pirzada* | ||
| Department of Mathematics, Hazratbal | ||
| چکیده | ||
| 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 $v_{i}$. | ||
| کلیدواژهها | ||
| Adjacency matrix؛ generalized adjacency matrix؛ spectral radius؛ clique number | ||
| مراجع | ||
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[1] A.E. Brouwer and W.H. Haemers, Spectra of Graphs, Springer, New York, 2012.
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[3] D. Li, Y. Chen, and J. Meng, The Aα-spectral radius of trees and unicyclic graphs with given degree sequence, Appl. Math. Comput. 363 (2019), Article ID: 124622.
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[5] T.S. Motzkin and E.G. Straus, Maxima for graphs and a new proof of a theorem of Turán, Canad. J. Math. 17 (1965), 533–540.
[6] V. Nikiforov, Merging the A and Q spectral theories, Appl. Analysis Discrete Math. 11 (2017), no. 1, 81–107.
[7] S. Pirzada, An Introduction to Graph Theory, Universities Press Orient BlackSwan, Hyderabad, 2012.
[8] S. Wang, D. Wong, and F. Tian, Bounds for the largest and the smallest Aαeigenvalues of a graph in terms of vertex degrees, Linear Algebra Appl. 590 (2020), 210–223. | ||
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