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Sharp Bounds on the Aα-index of Graphs in Terms of the Independence Number

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Abstract

Given a graph G, the adjacency matrix and degree diagonal matrix of G are denoted by A(G) and D(G), respectively. In 2017, Nikiforov[24] proposed the Aα-matrix: Aα(G) = αD(G) + (1 − α)A(G), where α ∈ [0, 1]. The largest eigenvalue of this novel matrix is called the Aα-index of G. In this paper, we characterize the graphs with minimum Aα-index among n-vertex graphs with independence number i for α ∈ [0, 1), where \(i = 1,\,\,\left\lfloor {{n \over 2}} \right\rfloor,\left\lceil {{n \over 2}} \right\rceil,\,\left\lfloor {{n \over 2}} \right\rfloor + 1,n - 3,n - 2,n - 1\), whereas for i = 2 we consider the same problem for \(\alpha \in [0,{3 \over 4}]\). Furthermore, we determine the unique graph (resp. tree) on n vertices with given independence number having the maximum Aα-index with α ∈ [0, 1), whereas for the n-vertex bipartite graphs with given independence number, we characterize the unique graph having the maximum Aα-index with \(\alpha \in [{1 \over 2},1)\).

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Acknowledgement

We take this opportunity to thank the anonymous reviewers for their critical reading of the manuscript and suggestions which have immensely helped us in getting the article to its present form.

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Authors and Affiliations

  1. Faculty of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China

    Wan-ting Sun & Shu-chao Li

  2. School of Information Eigeneering, Wuhan Business University, Wuhan, 430056, China

    Li-xia Yan

  3. The University of Georgia, Athens, GA, 30602, United States

    Xue-chao Li

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  1. Wan-ting Sun
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Correspondence to Shu-chao Li.

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The authors declare no conflict of interest.

This paper is supported by the Undergraduate Innovation and Entrepreneurship Grant from Central China Normal University (Grant No. 20210409037), by Industry-University-Research Innovation Funding of Chinese University (Grant No. 2019ITA03033) and by the National Natural Science Foundation of China (Grant Nos. 12171190, 11671164).

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Sun, Wt., Yan, Lx., Li, Sc. et al. Sharp Bounds on the Aα-index of Graphs in Terms of the Independence Number. Acta Math. Appl. Sin. Engl. Ser. 39, 656–674 (2023). https://doi.org/10.1007/s10255-023-1049-4

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  • DOI: https://doi.org/10.1007/s10255-023-1049-4

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