**Abstract:**
In this paper we study some concepts involving distance in subdivision graphs and inserted graphs and their centers. We prove some results on center, periphery and radius of the subdivision graph S (G) and inserted graph I(G)of a graph G for complete, cycle, complete bipartite, star and wheel graphs. An important theorem has been proved that in a connected graph the centers of G and S (G) have no common vertex. Graphs which are the periphery of some subdivision graph are characterized.

**Keywords:**
Distance, Eccentricity, Radius, Diameter, Center, Subdivision graph and Inserted graph.