vertex that can reach every other vertex in the graph, imply every vertex can reach each other? Given an undirected graph G.
If there is a vertex that can reach every other vertex in the graph, does that imply, every other vertex can reach each other?

Yes, to see this calculate the square of the adjacency matrix of the graph, $A^2$. You'll recognize that every matrix element is $>0$, indicating that there is at least one 2-step-path from every vertex to any other...
