Given a connected \((p, q)\) graph with a number of central vertices, form a new graph \(G^*\) as follows: \(V(G^*) = V(G)\); Delete all the edges of \(G\). Introduce an edge between every central vertex to each and every non-central vertex of \(G\); allow every pair of central vertices to be adjacent. In this paper, we probed \(G^*\) and deduced a number of results.