Extremal Arithmetic-Geometric Index of Bicyclic Graphs

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) = Sigma(xy is an element of E(G)) d(x)+d(y)/2 root dx d(y), where d(x) is the degree of vertex x. Vuki ' cevi ' c 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+ v(10)/(root 6) <= AG(G) <= (n+2)/(2 root 3(n-1)) + (n+1)/(root 2(n-1)) + (n(n-4))/(2 root n-1) + (5)/(root 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.