SUMMARY
In a uniform central graph (UCG) the set of eccentric vertices of a central vertex is the same for all central vertices. This collection of eccentric vertices is the centered periphery. For a pair of graphs (C,P) the central-peripheral appendage number, Aucg(C,P), is the minimum number vertices needed to be adjoined to the graphs C and P in order to construct a uniform central graph H with center V(C) and centered-periphery V(P). We compute Aucg(C,P) in terms of the radius and diameter of P and whether or not C is a complete graph. In the process we show Aucg(C, P) = 6 if diam(P) > 2. We also provide structure theorems for UCGs in terms of the centered periphery.