A class of additive multiplicative graph functions (Q1103643)

For a fixed graph G, the capacity function for G, \(P_ G\), is defined by \(P_ G(H)=\lim_{n\to \infty}[\nu_ G(H\quad n)]^{1/n},\) where \(\nu_ G(H)\) is the maximum number of disjoint G's in H. Hsu proved that \(P_{K_ 2}\) can be viewed as a lower bound for multiplicative increasing graph functions. But it was not known whether \(P_{K_ 2}\) is multiplicative or not. In this paper, they prove that \(P_ G\) is multiplicative and additive for some graphs G which include \(K_ 2\). Some properties of \(P_ G\) are also discussed in this paper.