Strong Independence of Graphcopy Functions

Let H be a finite graph on v vertices . We define a function CH , with domain the set of all finite graphs, by letting cH(G) denote the fraction of subgraphs of G on v vertices isomorphic to H. Our primary aim is to investigate the behavior of the functions CH with respect to each other. We show that the CH, where H is restricted to be connected, are independent in a strong sense. We also show that, in an asymptotic sense, the cH, H disconnected, may be expressed in terms of the CH , H connected . In 1932, Whitney [1] proved that the functions CH , H connected, were algebraically independent . Our results may be considered an extension of this work .