On atom–bond connectivity index of connected graphs

The atom–bond connectivity (ABC) index of a graph GG is defined as ABC(G)=∑uv∈E(G)du+dv−2dudv, where E(G)E(G) is the edge set and dudu is the degree of vertex uu of GG. 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 nn-vertex unicyclic graphs with the maximum, the second, the third and the fourth maximum ABC indices, and the nn-vertex bicyclic graphs with the maximum and the second maximum ABC indices respectively for n≥5n≥5.

► We give upper bounds for the atom–bond connectivity (ABC) index of connected graphs.
► We determine the unicyclic graphs with first four largest ABC indices for a given order.
► We determine the bicyclic graphs with first two largest ABC indices for a given order.