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Extremal Results for Cacti

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Abstract

In this paper, we determine the unique maximum (or minimum) extremal graph for general spectral radius, zeroth-order general Randić index, general sum-connectivity index, general Platt index, second Zagreb index and multiplicative Zagreb indices in the class of cacti with \(n\ge 4\) vertices and \(k\ge 0\) cycles. By applying our new result, we demonstrate the unique maximum extremal graph for general spectral radius, zeroth-order general Randić index, general sum-connectivity index, general Platt index, second Zagreb index and second multiplicative Zagreb index in the class of cacti with \(n\ge 4\) vertices. Furthermore, we also determine the unified maximum extremal graphs for general spectral radius, zeroth-order general Randić index and the second multiplicative Zagreb index in the class of cacti with \(n\ge 4\) vertices and \(k\ge 1\) pendant vertices.

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Acknowledgements

The authors would like to thank the referees for their valuable comments which lead to an improvement in the original manuscript. The first author is partially supported by NNSF of China (No. 11571123), Guangdong Province Ordinary University Characteristic Innovation Project (No. 2017KTSCX020). The third author is supported by the National Research Foundation of the Korean government with Grant No. 2017R1D1A1B03028642.

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Authors and Affiliations

  1. Department of Mathematics, South China Agricultural University, Guangzhou, 510642, People’s Republic of China

    Muhuo Liu & Yuedan Yao

  2. Department of Mathematics, Sungkyunkwan University, Suwon, 16419, Republic of Korea

    Kinkar Chandra Das

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  1. Muhuo Liu
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  2. Yuedan Yao
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  3. Kinkar Chandra Das
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Correspondence to Kinkar Chandra Das.

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Communicated by Xueliang Li.

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Liu, M., Yao, Y. & Das, K.C. Extremal Results for Cacti. Bull. Malays. Math. Sci. Soc. 43, 2783–2798 (2020). https://doi.org/10.1007/s40840-019-00837-2

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  • DOI: https://doi.org/10.1007/s40840-019-00837-2

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