On atom–bond connectivity index of connected graphs

Abstract

The atom–bond connectivity (ABC) index of a graph G is defined as A B C ( G ) = ∑ u v ∈ E ( G ) d u + d v − 2 d u d v , where E ( G ) is the edge set and d u is the degree of vertex u of G . We give an upper bound for the ABC index of connected graphs with fixed number of vertices, number of edges and maximum degree, and characterize the extremal graphs. From this, we obtain an upper bound and extremal graphs for the ABC index of molecular graphs with fixed number of vertices and number of edges. Then we determine the n -vertex unicyclic graphs with the maximum, the second, the third and the fourth maximum ABC indices, and the n -vertex bicyclic graphs with the maximum and the second maximum ABC indices respectively for n ≥ 5 .