Enumeration of spanning subgraphs with degree constraints

Details Enumeration of spanning subgraphs with degree constraints Enumeration of spanning subgraphs with degree constraints

2004-12-02

arxiv.org/abs/math/0412059v2

Mathematics

Combinatorics

15 pages. minor corrections and a new result

Scientific paper

For a finite undirected multigraph G=(V,E) and functions f,g:V-->\NN, let N_f^g(G,j) denote the number of (f,g)-factors of G with exactly j edges. The Heilmann-Lieb Theorem implies that \sum_j N_0^1(G,j) t^j is a polynomial with only real (negative) zeros, and hence that the sequence {N_0^1(G,j)} is strictly logarithmically concave. Separate generalizations of this theorem were obtained by Ruelle and by the author. We unify, simplify, and generalize these results by means of the Grace-Szeg\"o-Walsh Coincidence Theorem.

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