In this brief paper, using results from the theory of averaging of bilinear systems, we provide a graph theoretic characterization for the existence of periodic control inputs that stabilize a sparse matrix system to the origin. In particular, we introduce a class of extensions to the directed graph corresponding to a given sparse matrix system, which when contains a stable sparse matrix system implies that the original system is stabilizable using periodic inputs. When this condition holds, we provide a systematic procedure for designing such controllers. Our technical approach combines ideas from the theory of bilinear control systems and averaging theory with graph theoretic results. (C) 2016 Elsevier Ltd. All rights reserved.