Abstract
Let \(G\) be a connected graph with vertex set \(V(G)\). The multiplicatively weighted Harary index of a graph \(G\) is defined as \(H_M(G)=\sum _{u\ne v}\frac{d(u)d(v)}{d(u,v)}\), where \(d(u)\) is the degree of vertex \(u\), and \(d(u,v)\) denotes the distance between \(u\) and \(v\). In this paper, we first prove that the multiplicatively weighted Harary index of a graph is monotonic on some transformations, and then determine the extremal values of the multiplicatively weighted Harary indices for some familiar classes of graphs and characterize the corresponding extremal graphs.
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Acknowledgments
The authors are grateful to the referees for their valuable comments on this paper. The first author’s research is supported by Hunan Provincial Natural Science Foundation of China (13JJ3053).
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Deng, H., Krishnakumari, B., Venkatakrishnan, Y.B. et al. Multiplicatively weighted Harary index of graphs. J Comb Optim 30, 1125–1137 (2015). https://doi.org/10.1007/s10878-013-9698-5
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DOI: https://doi.org/10.1007/s10878-013-9698-5