Maximal Unicyclic Graphs With Respect to New Atom-bond Connectivity Index

摘要

The concept of atom-bond connectivity (ABC) index was introduced in the chemical graph theory in 1998. The atom-bond connectivity (ABC) index of a graph G defined as ABC(G) = ∑from v_iv_j∈E(G) √((d_i)+d_j-2)/(d_id_j) where E(G) is the edge set and d_i is the degree of vertex v_i of G. Very recently Graovac et al. define a new version of the ABC index as ABC2(G) = ∑from v_iv_j∈E(G) √((n_i)+n_j-2)(_in_j) where n_i denotes the number of vertices of G whose distances to vertex v_i are smaller than those to the other vertex v. of the edge e = v_i v_j, and n_j is defined analogously. In this paper we determine the maximal unicyclic graphs with respect to new atom-bond connectivity index (ABC2).