In this work, we improve the efficiency of Unit Commitment (UC) optimization solvers using a Graph Convolutional Neural Network (GCNN). In power systems, UC is crucial as it entails making essential decisions regarding the scheduling of power generation units to effectively meet the demand within a specific period. The complex nature of UC, arising from the binary nature of the problem and the non-convexity of the power flow constraints, warrants their representation as mixed-integer second-order cone program (MISOCP). Harnessing the inherent structure of these mixed-integer programs, we represent them as variable-constraint k-partite graphs. Utilizing a GCNN, we extract valuable insights from these graphs, learning effective variable selection policies for the branch and bound algorithm, thereby accelerating the overall optimization process. Our method ensures the preservation of solution optimality contrary to end-to-end learning approaches. Our methodology unfolds in two parts: (1) we construct a k-partite graph from the constraints of the optimization problem (2) we subsequently train our GCNN model on this graph representation via imitation learning, using the strong branching expert rule as our guide. Our approach stands out by effectively integrating problem-specific information into the variable selection within the branch and bound process of optimization.