منابع مشابه
Counting rooted forests in a network
If F,G are two n×m matrices, then det(1+xFG) = ∑ P x |P det(FP )det(GP ) where the sum is over all minors [19]. An application is a new proof of the Chebotarev-Shamis forest theorem telling that det(1 + L) is the number of rooted spanning forests in a finite simple graph G with Laplacian L. We can generalize this and show that det(1 + kL) is the number of rooted edge-k-colored spanning forests....
متن کاملCounting forests by descents and leaves
A descent of a rooted tree with totally ordered vertices is a vertex that is greater than at least one of its children. A leaf is a vertex with no children. We show that the number of forests of rooted trees on a given vertex set with i+1 leaves and j descents is equal to the number with j+1 leaves and i descents. We do this by finding a functional equation for the corresponding exponential gen...
متن کاملCounting rooted spanning forests in complete multipartite graphs
Jin and Liu discovered an elegant formula for the number of rooted spanning forests in the complete bipartite graph a1;a2 , with b1 roots in the rst vertex class and b2 roots in the second vertex class. We give a simple proof to their formula, and a generalization for complete m-partite graphs, using the multivariate Lagrange inverse. Y. Jin and C. Liu [3] give a formula for f(m; l;n; k), the n...
متن کاملParameterized Counting of Trees, Forests and Matroid Bases
We investigate the complexity of counting trees, forests and bases of matroids from a parameterized point of view. It turns out that the problems of computing the number of trees and forests with k edges are #W[1]-hard when parameterized by k. Together with the recent algorithm for deterministic matrix truncation by Lokshtanov et al. (ICALP 2015), the hardness result for k-forests implies #W[1]...
متن کاملA Randomised Approximation Algorithm for Counting the Number of Forests in Dense Graphs
r(A) = \V\-k(A), (2) where k(A) is the number of connected components of the graph with vertex set V and edge set A (including isolated vertices). It can immediately be seen from the above that the number of forests of a graph is equal to the value of the Tutte polynomial at the point (2,1). The Tutte polynomial contains many other invariants of fundamental importance in fields as diverse as st...
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ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 1990
ISSN: 0012-365X
DOI: 10.1016/0012-365x(90)90139-9