All Questions
Tagged with spanning-tree reference-request
7 questions
1
vote
0
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378
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Counting number of spanning trees of the complete bipartite with given vertex-degrees
For given $n_1,n_2 \in \mathbb{N}$ let $K_{n_1,n_2}$ be the complete bipartite graph. I have seen a few sources proving that the number of spanning trees $t(K_{n_1,n_2})$ is given by $n_1^{n_2-1} n_2^{...
- 1,295
2
votes
1
answer
254
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Is there a formula for the number of trees with this extra condition?
A tree $G$ on $n$ vertices $V=\{v_1,...,v_n\}$ is a connected undirected graph which is acyclic. For each tree $G$ one can split the set of vertices $V$ into two disjoint subsets $U,W \subset V$ such ...
- 1,295
7
votes
0
answers
171
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What is known about the distribution of lengths of the cycle you get by adding an edge to a uniform spanning tree?
Let $G$ be a finite, connected graph. Let $T$ be a uniform spanning tree, and let $e$ be a uniformly random edge not in $T$. When we add $e$ to $T$, we get a subgraph with a unique cycle, $C$. I am ...
- 1,462
5
votes
1
answer
281
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Transfer-impedance matrix for edge correlations in random spanning tree
Suppose $G$ is a (weighted) connected graph and
let $T$ denote a random spanning tree of $G$,
chosen uniformly (or respecting the edge weights).
It is known that for any distinct edges $e, f$
$$\...
- 2,175
10
votes
2
answers
1k
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History of deletion-contraction formula
The following is known as deletion-contraction formula:
Assume $\Gamma$ is a connectted graph with edge $\rho$ then
$$t(\Gamma)=t(\Gamma\backslash\rho)+t(\Gamma/\rho),$$
where $\Gamma\backslash\...
- 45k
6
votes
1
answer
643
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Random path in a graph
Consider a finite graph $G$. I would like to define a random path between two vertices $s$ and $t$ of the graph $G$ by looking at a measure $\mu$ on all spanning trees. Then the probability of a given ...
- 4,422
2
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0
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1k
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Incremental minimum spanning tree
Given a connected graph $G=(V,E)$ with a weight function $w:E\to\mathbb{R}$ and a subset $E_0\subseteq E$ such that the subgraph $(V,E_0)$ is connected, I am looking for a sequence $E_0\subseteq E_1\...
- 2,966