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 اصل مقاله (391.25 K)
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
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A note on the maximum $A_{\alpha}$-spectral radius of some classes of graphs