The harmonic index of a graph $G$ is defined as $ H(G)=sumlimits_{uvin E(G)}frac{2}{d(u)+d(v)}$‎, ‎where $d(u)$ denotes the degree of a vertex $u$ in $G$‎. ‎Let $mathcal{G}(n,k)$ be the set of simple $n$-vertex graphs with ...