Atom-Bond Connectivity Index of Unicyclic Graphs with Perfect Matchings

Abstract

The atom-bond connectivity (ABC) index of a graph \(G\) is defined in mathematical chemistry as\(\mathrm{ABC}(G) = \sum_{uv \in E(G)} \sqrt{\frac{d_u +d_v-2}{ d_u d_v}},\) where \(E(G)\) is the edge set of \(G\) and \(d_u\) is the degree of vertex \(u\) in \(G\).In this paper, we determine the unique graphs with the largest and the second largest ABC indices, respectively, in the class of unicyclic graphs on \(2m\) vertices with perfect matchings.