We present a work-efficient parallel level-synchronous Breadth First Search (BFS) algorithm for shared-memory architectures which achieves the theoretical lower bound on parallel running time. The optimality holds regardless of the shape of the graph. We also demonstrate the implication of this optimality for the energy consumption of the program empirically. The key idea is never to use more processing cores than necessary to complete the work in any computation step efficiently. We keep the rest of the cores idle to save energy and to reduce other resource contentions (e.g., bandwidth, shared caches, etc). Our BFS does not use locks and atomic instructions and is easily extendible to shared-memory coprocessors.