Asymptotic Enumeration of Spanning Trees

Asymptotic Enumeration of Spanning Trees

We give new formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay [Europ. J. Combin. 4 149–160] for regular graphs. The general answer involves a quantity for infinite graphs that we call ‘tree entropy’, which we show is a logarithm of a normalized determinant of the graph Laplacian for infinite graphs. Tree entropy is also exp...

Enumeration of Common Spanning Trees

Enumeration of spanning trees of certain graphs

In this note we give an algorithm which enables us to encode and enumerate all the spanning trees of a multipartite graph (see below). This algorithm may turn out to be useful for the enumeration of spanning trees satisfying certain conditions. The number of spanning trees of a given graph Γ without loops and without multiple edges will be denoted by_i(T). We shall consider the graphs Γ = T(G; ...

Enumeration of Spanning Trees of Graphs

In this paper, we define and study polynomials that enumerate spanning trees according to degrees of all vertices. These polynomials have a certain reciprocity. A formula that reduces the polynomials for multipartite graphs to the polynomials for component graphs is also found. There is a generalization of Prüfer’s coding that corresponds to the latter formula. All results are extended to the c...

Enumeration of spanning trees in simplicial complexes

The Kirchoff Matrix Tree Theorem states that the number of spanning trees in a graph G is equal to the absolute value of any cofactor of the Laplacian matrix of G. As the theory of simplicial complexes is a generalization of the theory of graphs one would suspect that there is a generalization of the notion of spanning trees to simplicial complexes, such that the number of spanning trees in a g...