Entropy of the weighted graphs with the ABC index as weights

The atom-bond connectivity $(ABC)$ index of a graph $G$ is defined as $$ABC(G)=\sum_{uv \epsilon E(G)}\sqrt{\frac {d_{u}+d_{v}-2}{d_{u}d_{v}}},$$ where $E(G)$ is the edge set and $du$, $dv$ are, respectively, the degrees of the vertices $u$ and $v$ in $G$. In this paper, we present some sharp lower and upper bounds on $ABC$.