On a Vertex-Degree-Based Graph Invariant and the Multiplicatively Weighted Harary Index

Abstract

The multiplicatively weighted Harary index (\(Hy\)-index) is a new distance-based graph invariant, which was introduced and studied by Deng et al. in [1]. For a connected graph \(G\), the multiplicatively weighted Harary index of \(G\) is defined as \(H_M(G) = \sum\limits_{\{u,v\} \subseteq V(G)} \frac{d_G(u) \cdot d_G(v)}{d_G(u,v)}\), where \(d_G(x)\) denotes the degree of vertex \(x\) and \(d_G(s,t)\) denotes the distance between vertices \(s\) and \(t\) in \(G\). In this paper, we first study a new vertex degree-based graph invariant \(M_2 – \frac{1}{2}M_1\), where \(M_1\) and \(M_2\) are ordinary Zagreb indices. We characterize the trees attaining maximum value of \(M_2 – 4M_1\) among all trees of given order. As applications, we obtain a new proof of Deng et al.’s results on trees with extremal \(H_M\)-index among all trees of given order.