On the enumeration of certain weighted graphs

  • Authors:

  • Miklós Bóna;Hyeong-Kwan Ju;Ruriko Yoshida

  • Affiliations:

  • Department of Mathematics, University of Florida, Gainesville, FL 32611, USA;Department of Mathematics, Chonnam National University, Kwangju 500-757, Republic of Korea;Department of Statistics, University of Kentucky, Lexington, KY 40506-0027, USA

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2007

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Abstract

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.