TUYMAADA 2024 S8

A graph $G$ has $n$ vertices ( $n>1$ ). For each edge $e$ let $c(e)$ be the number of vertices of the largest complete subgraph containing $e$ . Prove that the inequality (the summation is over all edges of $G$ ): $$\sum_{e} \frac{c(e)}{c(e)-1}\le \frac{n^2}{2}.$$