Malaysian Journal of Mathematical Sciences, June 2026, Vol. 20, No. 2
Extremal Values of The Harmonic Index of Graphs
Xu, C., Li, G.
Corresponding Email: xuchunlei1981@sina.cn
Received date: 11 July 2025
Accepted date: 31 October 2025
Abstract:
For a graph \(\Omega\) with vertex set \(V(\Omega)\) and edge set \(E(\Omega)\), its harmonic index is defined as \(H(\Omega)=\displaystyle\sum_{uv\in E(\Omega)}\dfrac{2}{d_u+d_v}\), where \(d_u\) and \(d_v\) denote the degrees of vertices \(u\) and \(v\), respectively. The harmonic index captures various structural characteristics of a graph, such as connectivity and the uniformity of its vertex degree distribution, and finds application in many fields, including molecular structure analysis, network layout optimization, and information dissemination analysis. In this paper, we first determine the extremal values of the harmonic index for chemical graphs, and chemical trees. Moreover, using some relevant operations, we fully characterize the trees with branch vertices that achieve the extremal values of the harmonic index.
Keywords: extremal value; extremal graph; tree; harmonic index.
