Find a particular connected graph within a complete graph keeping all vertices
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.