The Weighted Connected Vertex Cover (W-CVC) problem is a fundamental optimization problem in graph theory, combining the constraints of vertex covering, connectivity, and weight minimization. Given a weighted graph, the objective is to determine a minimum-weight vertex subset that covers all edges while inducing a connected subgraph. This paper focuses on the derivation of W-CVC specifically on Butterfly Graphs, a class of structured interconnection networks widely used in parallel computing. By exploiting the recursive and symmetric properties of butterfly graphs, an explicit derivation of the optimal weighted connected vertex cover is presented. The study provides a systematic approach suitable for exam preparation, highlighting key steps, structural insights, and theoretical justification.