We enumerate weighted graphs with a certain upper bound condition. 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 if the given graph is a bipartite graph, then its generating function is of the form $\frac{ p(x)}{ (1-x)^{ m+1}}$, where $m$ is the number of vertices of the graph and $p(x)$ is a polynomial of degree at most $m$.
Categories: math.CO