Abstract
The Balaban index (also called the J index) of a connected graph G is a distance-based topological index, which has been successfully used in various QSAR and QSPR modeling. Although the index was introduced 30 years ago, there are few results on the asymptotic relations. In this paper, several asymptotic bounds on the Balaban indices of trees with diameters 3 and 4 are shown, respectively.
Highlights
All graphs considered in this paper are simple and undirected
1, μ + 1 uv∈E(G) σG(u)σG(v) where μ is the cyclomatic number and μ m − n + 1. e Balaban index of a connected graph G is a distance-based topological index, which has been successfully used in various QSAR and QSPR modeling [2, 3]
By comparing with the Wiener index regarding alkanes in [7], it was found that the Balaban index reduces the degeneracy of the latter index and provides much higher discriminating ability
Summary
All graphs considered in this paper are simple and undirected. Let G be a graph with its edge set E(G) and vertex set. E Balaban index [1] of a connected graph G (or the J index for short) is defined as m J(G). Several asymptotic bounds on the Balaban indices of trees with diameters 3 and 4 are shown, respectively. En, this tree is denoted by Tn(3, a, b), see Figure 1, which has n vertices, diameter 3, and satisfies that there are a pendent edges attached at one end-vertex of one edge and b pendent edges attached at the other end-vertex of the edge, where a ≥ 1, b ≥ 1, and a + b n − 2. E set of this kind of trees is denoted by Tn(3, a, b). E tree with order n and diameter 4 denoted by Tln(4, a1, a2, . E set of this kind of trees is denoted by Tln(4, a1, a2, .
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