All Questions
Tagged with spanning-tree trees
13 questions
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Two independent spanning trees of $2$-connected graph with $P_5$-free and $K_{1,3}$-free
I'm going to prove the following statement:
$G$ is a $P_5$-free and $K_{1,3}$-free graph with $\vert G \vert \geq 7$, and $G \notin \mathcal{K}$, then $G$ is $2$-connected graph if and only if $G$ ...
- 11
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Approximating all spanning trees with their sample
In a complete graph with $n$ vertices there are $n^{n-2}$ trees.
In my research I'm analyzing trees in the following way (each edge has a weight):
Get a tree.
Build a complete graph, by the following ...
- 49
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54
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Minimum spanning tree and projection
Let $G$ be a graph of $n$ arcs and let $x\in \mathbb{R}^n$. I want to compute the orthogonal projection of $x$ onto the set of radial graphs with $k$ roots contained in $G$ (or a forest with $k$ root) ...
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How can we hang the weighted trees so that vertices nearer to root (based on distances, not hop count) lie in upper levels?
I have a set of edge weighted trees, each tree rooted at some vertex. Consider these trees are hung from the roots and vertices are arranged in some levels. I wish to design an algorithm (...
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1
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171
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Counting spanning trees of $K_{b+1,w+1}$ with certain properties or calculating a combinatorial sum
For $b,w \geq 0$ let $K_{b+1,w+1}$ be the complete bipartite graph with vertices $a_1,...,a_{b+1}$ on the left hand side and $c_1,...,c_{w+1}$ on the right hand side. For given $1 \leq d \leq w$ and $...
- 1,295
2
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1
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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
2
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2
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551
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Two independent spanning trees of $2$-connected graph
I want to prove the following statement:
Let $u$ be a vertex in a $2$-connected graph $G$. Then $G$ has two spanning trees such that for every vertex $v$, the $u,v$-paths in the trees are independent....
- 341
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70
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(Weakly) connected sets with large (out-)boundary
Let $\Gamma=(V,E)$ be a connected undirected graph with n vertices such that every vertex has degree at least $4$. Now draw arrows on some of the edges, in such a way that the in-degree of every ...
- 20.1k
4
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Relation between Kirchhoff's Circuital law and Matrix tree Theorem
I'm not a professional mathematician, just an undergraduate student. I was reading Introduction to Graph Theory by West, I came over the topic which discuses the methods to find the spanning trees in ...
- 143
7
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Is there a natural relationship between OEIS A127670 and Cayley's tree formula?
I apologize in advance that this question must sound highly amateurish, but I am wondering if there is any connection between the formula https://oeis.org/A127670 , which counts the number of fixed $n$...
- 4,049
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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
2
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105
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Is there a name for this variant of the MST and the TSP?
Suppose I am given a weighted graph $G$ that contains a "start vertex" $v_0$, and my goal is to construct a set of paths that all originate at $v_0$ and touch all of the vertices of $G$, with as ...
- 4,049
3
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2
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287
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on counting the number of trees on Kn (case)
During my reasearch I have stumbled across a problem that can be presented in such way:
"How many are there spanning trees on Kn such that every tree contains v: deg(v) = k, for a given k"
The ...
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