On the atom-bond connectivity index and radius of connected graphs

Abstract

Let G be a connected graph with edge set E(G)$E (G)$. The atom-bond connectivity index (ABC index for short) is defined as ABC(G)=∑uv∈E(G)dG(u)+dG(v)−2dG(u)dG(v)$ABC (G) = \sum_{uv \in E (G)} \sqrt{ \frac{d_{G} (u)+d_{G} (v)-2}{d_{G} (u) d_{G} (v)} }$, where dG(u)$d_{G} (u)$ denotes the degree of vertex u in G. The research of ABC index of graphs is active these years, and it has found a lot of applications in a variety of fields. In this paper, we will focus on the relationship between ABC index and radius of connected graphs. In particular, we determine the upper and lower bounds of the difference between ABC index and radius of connected graphs.