Tutte polynomial of an Eulerian graph


主讲人:马俊 上海交通大学




内容先容:William Tutte is one of the founders of the modern graph theory. For every  undirected graph G, Tutte defined a polynomial TG(x,y) in two variables which  plays an important role in graph theory. It encodes information about subgraphs  of $G$. For example, for a connected graph G, TG(1, 1) is the number of spanning  trees of G, TG(2, 1) is the number of spanning forests of G, TG(1, 2) is the  number of connected spanning subgraphs of G, TG(2, 2) is the number of spanning  subgraphs of G. One has been looking for analogues of the Tutte polynomial for  digraphs for a long time. Recently, considering an Eulerian digraph and a  Chip-firing game on this digraph, Kevin Perrot and Swee Hong Chan gave  generalizations of the partial Tutte polynomial TG(1,y) from the point of view  of recurrent congurations of the Chip-ring game. In this talk, let D be an  weak-connected Eulerian digraph and v be a vertex of D. We will introduce two  polynomials and , which are defined on the set of v-sink subgaphs and the set of  acyclic v-sink subgaphs of D, respectively. We find that these two polynomials  have very good invariance properties. In particular, these two polynomials are  independent of the choice of the vertex v.

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