On the Minimum and Maximum Complementary Geometric-Arithmetic Index of Connected Graphs

Abstract

Consider a graph G whose collection of edges is represented by E. For any vertex w ∈ V(G), let dw denote the number of edges incident to w, referred to as its degree. The graph invariant known as the complementary geometric-arithmetic index associated with G is defined as  , provided that d_v ≤ d_u. This paper provides some bounds on this index. The graphs maximizing/minimizing this index among all fixed-order (molecular) trees are also characterized. Only regular graphs minimize cGA among connected n-order graphs for every n ≥ 3. A computer-based approach is employed to exhaustively search the graphs maximizing cGA among connected n-order graphs for 5 ≤ n ≤ 10. Based on these computational findings, a conjecture is proposed, and two structural properties of the extremal graphs are provided.