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$\mathrm{A}\mathrm{B}\mathrm{C}(G)=\sum _{uv\in E(G)}\sqrt{\frac{d_u+d_v-2}{d_u{d}_v}},$ 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.