A periodic graph models various natural and art- ficial periodic patterns with repetitions of a given static graph, and have vast applications in crystallography, scheduling, VLSI circuits and systems of uniform recurrence equations. This paper considers a graph Voronoi diagram for a given subset of vertices on a periodic graph. The simplest two-dimensional periodic graph is a square lattice, and the Voronoi diagram on the lattice geometrically corresponds to the L1 Voronoi diagram in the plane. We extend this geometric relation for the two- dimensional periodic graphs, including the honeycomb lattice, kagome lattice and two-dimensional periodic graph with small static graph. For these graphs, the graph Voronoi diagram can be represented implicitly by Voronoi diagrams with respect to appropriate convex-distance functions with extra elaborations.