Inequalities involving the harmonic-arithmetic index

期刊论文
发表日期：

2024年4月11日

查看全文

Akbar Ali,
Emina Milovanović,
Stefan Stankov,
Marjan Matejić,
Igor Milovanović

摘要

Let G be a simple graph with vertex set $V=\{v_{1},v_{2},\ldots ,v_{n}\}$ $V = {v_{1}, v_{2}, \dots, v_{n}}$ . The notion $i\sim j$ $i \sim j$ is used to indicate that the vertices $v_{i}$ $v_{i}$ and $v_{j}$ $v_{j}$ of G are adjacent. For a vertex $v_{i}\in V$ $v_{i} \in V$ , let $d_{i}$ $d_{i}$ be the degree of $v_{i}$ $v_{i}$ . The harmonic-arithmetic (HA) index of G is defined as $HA(G) =\sum _{i\sim j} 4d_id_j(d_i+d_j)^{-2}$ $H A (G) = \sum_{i \sim j} 4 d_{i} d_{j} {(d_{i} + d_{j})}^{- 2}$ . In this paper, a considerable number of inequalities involving the HA index and other topological indices are derived. For every obtained inequality, all the graphs that satisfy the equality case are also characterized.