Extremal Arithmetic–Geometric Index of Bicyclic Graphs

Abstract

Recently, a novel topological index of graphs, called arithmetic–geometric (AG) index, has been proposed to characterize the nature of chemical compounds or interconnection networks. The arithmetic–geometric index of the graph G is denoted as $AG(G)=\sum _{xy\in E(G)}\frac{d_x+d_y}{2\sqrt{d_xd_y}}$, where $d_x$ is the degree of vertex x. Vukićević et al. (Discrete Appl Math 302:67–75, 2021) determined the n-vertex unicyclic graphs with extremal values of AG index and proposed a conjecture about extremal values of AG index in the class of bicyclic graphs. In this work, we confirm this conjecture and show that if G is an n-vertex connected bicyclic graph, then $n-3+\frac{10}{\sqrt{6}}\le AG(G)\le \frac{n+2}{2\sqrt{3(n-1)}}+\frac{n+1}{\sqrt{2(n-1)}}+\frac{n(n-4)}{2\sqrt{n-1}}+\frac{5}{\sqrt{6}}$. The left equation holds for the following two types of graphs: the graph consisting of two vertex-disjoint cycles $C_a$ and $C_b$ with $n=a+b$ connected by an edge and the graph derived by adding an edge between two non-adjacent vertices in $C_n$. The right equation holds for the graph constructed by adding $n-4$ pendant vertices to the vertex of degree 3 in $C_4^+$, where $C_4^+$ is the bicyclic graph derived from adding an edge between two non-adjacent vertices in $C_4$. Furthermore, we investigate the correlations of AG index with physico-chemical properties of octane isomers and show that it is a better predictor of molecular properties.