On the Balaban Index of Chain Graphs

The Balaban index and sum-Balaban index of a connected (molecular) graph G are defined as J(G)=mμ+1∑uv∈E(G)1σG(u)σG(v)andSJ(G)=mμ+1∑uv∈E(G)1σG(u)+σG(v),respectively, where m is the number of edges, μ is the cyclomatic number, σ_G(u) is the sum of distances between vertex u and all other vertices of G. In this paper, we establish that K(DS(n-3,1))>K(DS(n-4,2))>⋯>K(DS(⌈n2⌉-1,⌊n2⌋-1))(K=J,SJ), where DS(p,q) is a double star on n(=p+q+2,p≥q) vertices. As an application, we determine the extremal graphs of the Balaban index and the sum-Balaban index in the class of chain graphs G on n vertices, where G is a tree or a unicyclic graph. Finally, we give an open problem on Balaban (sum-Balaban) index of connected chain graphs.