On the enumeration of certain weighted graphs |
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Authors: | Miklós Bóna Ruriko Yoshida |
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Institution: | a Department of Mathematics, University of Florida, Gainesville, FL 32611, USA b Department of Mathematics, Chonnam National University, Kwangju 500-757, Republic of Korea c Department of Statistics, University of Kentucky, Lexington, KY 40506-0027, USA |
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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 p1(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)l. Here m is the number of vertices of the graph, p1(x) is a symmetric polynomial of degree at most m, p2(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. |
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Keywords: | Weighted graphs Rational convex polytopes Rational generating functions Ehrhart (quasi-)polynomial |
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