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On reduced second Zagreb index

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Abstract

The reduced second Zagreb index \(RM_2\) of a graph G is defined as \(RM_2(G)=\sum _{uv\in E(G)}(d_G(u)-1)(d_G(v)-1)\), where \(d_G(u)\) is the degree of the vertex u of graph G. Furtula et al. (Discrete Appl Math 178: 83–88, 2014) studied the difference between the classical Zagreb indices of graphs and showed that it is closely related to \(RM_2\). In this paper, we obtain an upper bound in terms of order n and size m on \(RM_2\) of \(K_{r+1}\)-free graphs. Also we prove that among all graphs of order n with chromatic number \(\chi \), the Turán graph \(T_{n,\,\chi }\) is the unique graph having the maximum \(RM_2\). Furthermore, we completely characterize the extremal graphs with respect to \(RM_2\) among all unicyclic graphs of order n with girth g.

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Acknowledgements

This work has been done within the framework of the project P2019-3640 (ARC 2019-2020) supported by the Asia Research Center, Mongolia and Korea Foundation for Advanced Studies, Korea. K. C. Das is supported by the National Research Foundation of the Korean government with Grant No. 2017R1D1A1B03028642.

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Correspondence to Kinkar Chandra Das.

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Buyantogtokh, L., Horoldagva, B. & Das, K.C. On reduced second Zagreb index. J Comb Optim 39, 776–791 (2020). https://doi.org/10.1007/s10878-019-00518-7

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