Generalized homomorphism graph functions

AbstractA real-valued function ƒ defined on the set of all graphs, G, such that ƒ(G×H)=ƒ(G)ƒ(H) for all G, H ϵ G is called multiplicative; and ƒ(G)⩽ƒ(H) whenever 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, ϕG, S for all G ϵ G and ∅≠S⊆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