A note on the maximum $A_{\alpha}$-spectral radius of some classes of graphs

Kalita, Jharna; Paul, Somnath; G, Indulal; Scaria, Deena C

doi:10.22049/cco.2025.30097.2308

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	A note on the maximum $A_{\alpha}$-spectral radius of some classes of graphs
Communications in Combinatorics and Optimization
مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 21 خرداد 1404
نوع مقاله: Original paper
شناسه دیجیتال (DOI): 10.22049/cco.2025.30097.2308
نویسندگان
Jharna Kalita¹؛ Somnath Paul¹؛ Indulal G^* ²؛ Deena C Scaria²^{، 3}
¹Department of Applied Sciences, Tezpur University, Napaam-784028, Assam, India
²Department of Mathematics, St Aloysius College, Edathua, Alappuzha, 689573. India
³Department of Mathematics, Marthoma College, Thiruvalla, India
چکیده
According to Nikiforov [V. Nikiforov, Merging the A- and Q-spectral theories, Appl. Anal. Discrete Math. 11 (2017), no. 1, 81–107], the $A_\alpha$-matrix of a graph $G$ is defined as $A_\alpha(G) = \alpha D(G) + (1-\alpha)A(G)$, where $\alpha \in [0, 1]$, $D(G)$ is the diagonal matrix with the degrees of the vertices of $G$ as the diagonal entries, and $A(G)$ is the adjacency matrix. The $A_\alpha$-spectral radius of the $A_\alpha$-matrix is its largest eigenvalue. In this study, we characterize the graph that maximizes the $A_\alpha$-spectral radius within three specific classes of graphs: (i) graphs of order $n$, with vertex connectivity $\kappa(G) \leq k$ and minimum degree $\delta(G) \geq k$; (ii) bipartite graphs of order $n$ with vertex connectivity $k$; and (iii) graphs of order $n$, connectivity $k$, and independence number $r$. Furthermore, for each of these three families, we determine the location of the $A_\alpha$-spectral radius.
کلیدواژه‌ها
Aα-matrix؛ Aα-spectral radius؛ vertex connectivity؛ minimum degree؛ bipartite graphs

مراجع
[1] A. Berman and R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM: Philadelphia, PA, USA, 1994. [2] H. Guo and B. Zhou, On the α-spectral radius of graphs, Appl. Anal. Discrete Math. 14 (2020), no. 2, 431–458. [3] W.H. Haemers, Interlacing eigenvalues and graphs, Linear Algebra Appl. 226 (1995), 593–616. https://doi.org/10.1016/0024-3795(95)00199-2 [4] D. Li, Y. Chen, and J. Meng, The aα-spectral radius of graphs with given degree sequence, arXiv preprint arXiv:1806.02603 (2018) [5] J. Li, W.C. Shiu, W.H. Chan, and A. Chang, On the spectral radius of graphs with connectivity at most k, J. Math. Chem. 46 (2009), no. 2, 340–346. https://doi.org/10.1007/s10910-008-9465-5 [6] H. Lin, X. Huang, and J. Xue, A note on the $A_{\alpha}$-spectral radius of graphs, Linear Algebra Appl. 557 (2018), 430–437. https://doi.org/10.1016/j.laa.2018.08.008 [7] X. Liu and S. Liu, On the $A_{\alpha}$-characteristic polynomial of a graph, Linear Algebra Appl. 546 (2018), 274–288. https://doi.org/10.1016/j.laa.2018.02.014 [8] H. Lu and Y. Lin, Maximum spectral radius of graphs with given connectivity, minimum degree and independence number, J. Discrete Algorithms 31 (2015), 113–119. https://doi.org/10.1016/j.jda.2014.08.006 [9] V. Nikiforov, Merging the A- and Q-spectral theories, Appl. Anal. Discrete Math. 11 (2017), no. 1, 81–107. [10] V. Nikiforov, G. Pastén, O. Rojo, and R.L. Soto, On the $A_{\alpha}$-spectra of trees, Linear Algebra Appl. 520 (2017), 286–305. https://doi.org/10.1016/j.laa.2017.01.029 [11] V. Nikiforov and O. Rojo, On the $\alpha$-index of graphs with pendent paths, Linear Algebra Appl. 550 (2018), 87–104. https://doi.org/10.1016/j.laa.2018.03.036 [12] C. Wang and S. Wang, The $A_{\alpha}$-spectral radii of graphs with given connectivity, Mathematics 7 (2019), no. 1, Article ID: 44. https://doi.org/10.3390/math7010044 [13] J. Xue, H. H. Lin, S. Liu, and J. Shu, On the $A_{\alpha}$-spectral radius of a graph, Linear Algebra Appl. 550 (2018), 105–120. https://doi.org/10.1016/j.laa.2018.03.038
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A note on the maximum $A_{\alpha}$-spectral radius of some classes of graphs