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 p1(x)/(1-x)m+1p1(x)/(1-x)m+1. For nonbipartite simple graphs, we get a generating function of the form p2(x)/(1-x)m+1(1+x)lp2(x)/(1-x)m+1(1+x)l. Here m   is the number of vertices of the graph, p1(x)p1(x) is a symmetric polynomial of degree at most m  , p2(x)p2(x) is a polynomial of degree at most m+lm+l, and l is a nonnegative integer. In addition, we give computational results for various graphs.