[math-fun] Graph question

Suppose a graph G means: * set V of points called vertices; * a collection E of unordered pairs of vertices called edges; * any two vertices have a path of edges connecting them: G is connected. (No edges from a vertex to itself, and no multiple edges between two vertices.) A subgraph H of G is a graph H whose vertices and edges belong to G. Call a graph G "homogeneous" if every isomorphism between subgraphs of G extends to an isomorphism of G with itself. Question: --------- Given a pair of positive integers (n,k), when does there exist a homogeneous graph G such that * every vertex has valence n, and * every vertex has just k vertices that are not adjacent to it ??? —Dan