On the enumeration of certain weighted graphs

We enumerate weighted simple graphs with a natural upper bound condition on the sum of the weight of adjacent vertices. We also compute the generating function of the numbers of these graphs, and prove that it is a rational function. In particular, we show that the generating function for connected bipartite simple graphs is of the form p"1(x)/(1-x)^m^+^1. For nonbipartite simple graphs, we get a generating function of the form p"2(x)/(1-x)^m^+^1(1+x)^l. Here m is the number of vertices of the graph, p"1(x) is a symmetric polynomial of degree at most m, p"2(x) is a polynomial of degree at most m+l, and l is a nonnegative integer. In addition, we give computational results for various graphs.