Find a particular connected graph within a complete graph keeping all vertices My problem is as follow, imagine a complete undirected graph with weighted edges, i would like to extract a connected graph keeping all the vertices and where the edge with the highest weight is the minimum one compared to all others possible configurations.

Is there another solution than bruteforce it.
Clearly you can focus on trees, since removing edges can only reduce the weight of the maximal edge. What you're looking for is called a minimum bottleneck spanning tree, and an MST is also a minimum bottleneck spanning tree.

To see why, suppose for the purpose of contradiction that $T$ is an MST which does not minmize the maximum weight. Let $(u,v)$ be the maximal weight edge in $T$, and $T'$ be a minimum bottleneck spanning tree. By our assumption $(u,v)\notin T'$, hence there exists a path from $u$ to $v$ in $T'$ which does not contain $(u,v)$, let us denote this path by $P$. Note that all the edges in $P$ are lighter than $(u,v)$. Remove $(u,v)$ from $T$, now $u,v$ are in different connected components. Suppose $P=(w_1=u,...,w_k=v)$, and denote by $i$ the minimal index such that $w_i$ and $w_{i+1}$ are in different connected components in $T\setminus\{(u,v)\}$. Now, $\left(T\setminus\{(u,v)\}\right)\cup\{(w_i,w_{i+1})\}$ is a spanning tree of $G$ with smaller weight, contradicting the fact that $T$ is an MST.