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

Articles in Press

Document Type : Original paper

Authors

1 Department of Applied Sciences, Tezpur University, Napaam-784028, Assam, India

2 Department of Mathematics, St Aloysius College, Edathua, Alappuzha, 689573. India

3 Department of Mathematics, Marthoma College, Thiruvalla, India

Abstract

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.

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References

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Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 11 June 2025

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