Generalized homomorphism graph functions

A real-valued function f defined on the set of all graphs, 3, such that f (G x H) = f (C)f (H) for all G, HE 52 is called multiplicative; and f(G) <f(H) w h enever G is a subgraph of H is called increasing. The classification of multiplicative increasing graph functions is still open. Up to now, there are a lot of known multiplicative increasing graph functions. In this paper, we introduce a new class of multiplicative increasing graph functions, namely, ~)o,~ for all G E % and 0 # S E V(G), defined to be the number of all possible homomorphic images of S for the homomorphism from G into H. Several properties of additive multiplicative increasing graph functions are also discussed in this paper.