On the enumeration of certain weighted graphs

@article{Bna2007OnTE,
  title={On the enumeration of certain weighted graphs},
  author={M. B{\'o}na and Hyeong-Kwan Ju and R. Yoshida},
  journal={Discret. Appl. Math.},
  year={2007},
  volume={155},
  pages={1481-1496}
}
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… Expand
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