Can two or more strongly connected components of a graph share a common vertex? I tried few examples but could not arrive at a formal proof that whether two or more strongly connected components of a graph share a common vertex. Does it have any formal proof? 

Strongly connected components of a graph are pairwise disjoint.
You may define the relation $u\sim v$ if and only if there is a $(u,v)$-path and a $(v,u)$-path in the graph. It is not hard to prove that this is an equivalence relation. Thus, the equivalence classes partition the set of vertices and each equivalence class induces a strongly connected component in the graph.
