10.1007/s40096-019-0287-3

The harmonic index of graphs based on some operations related to the lexicographic product

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  • B. N. Onagh*1,
  1. Department of Mathematics, Faculty of Sciences, Golestan University, Gorgan, IR
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Published in Issue 2019-06-08

How to Cite

Onagh, B. N. (2019). The harmonic index of graphs based on some operations related to the lexicographic product. Mathematical Sciences, 13(2 (June 2019). https://doi.org/10.1007/s40096-019-0287-3
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Abstract

Abstract The harmonic index of a graph G is defined as the sum of the weights 2degG(u)+degG(v)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{2}{\deg _G(u)+\deg _G(v)}$$\end{document} of all edges uv of G , where degG(u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\deg _G(u)$$\end{document} denotes the degree of a vertex u in G . In this paper, we investigate the harmonic index of graphs based on operations related to the lexicographic product, subdivision graph, t -subdivision graph, vertex-semitotal graph, edge-semitotal graph and total graph.

Keywords

  • Harmonic index,
  • Subdivision,
  • Lexicographic product,
  • F-product
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