Abstract
In this paper, we consider an invariant \(I(G)\) of a graph \(G=(V,E)\) defined as a summation over all edges, \(I(G) = \sum {c_{ij}x_{ij}}\) where \(c_{ij}\) and \(x_{ij}\) is the weight and number, respectively, of edges in \(G\) connecting vertices of degree \(i\) and \(j\). The graph invariant \(I(G)\) unifies Randić index, Zagreb index, sum–connectivity index, \(GA_1\) index, ABC index and harmonic index. Based on linear programming methods, we give the extremal values and extremal graphs of \(I(G)\) among all simple graphs of order \(n\) without isolated vertices. Applying this result, we obtain some extremal values of the Randić, Zagreb, sum–connectivity, \(GA_1\), ABC, and harmonic indices along with the corresponding graphs that obtain these values.
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Acknowledgments
We wish to thank the referees for their valuable comments and suggestions, which led to an improvement of the original manuscript. Project supported by Hunan Provincial Natural Science Foundation of China (13JJ3053)
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Deng, H., Huang, G. & Jiang, X. A unified linear-programming modeling of some topological indices. J Comb Optim 30, 826–837 (2015). https://doi.org/10.1007/s10878-013-9672-2
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DOI: https://doi.org/10.1007/s10878-013-9672-2