All Questions
Tagged with spanning-tree graph-theory
48 questions
2
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48
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On planar graphs with specific spanning tree count and poly number of vertices
Given set $\mathcal T_n=\{0,1,3,4\dots,2^n-1\}$ (note there is no $2$) what is the minimum number of vertices $m$ needed in a planar graph such that at every $i\in\mathcal T_n$ there is a graph $G\in\...
- 13.9k
2
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1
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105
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"Spanning trees" for connected linear hypergraphs
Starting point. For every simple, undirected graph $G=(V,E)$ there is $E_0\subseteq E$ such that $(V,E_0)$ is minimally connected: the spanning tree. The goal of this question is to find out whether ...
- 48.8k
1
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38
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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 $G$ ...
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10
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1
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901
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Could someone explain the proof of this formula clearly? I got the wrong values for spanning trees with this formula and with Cayley's formula
The passage quoted below is from "The number of spanning trees of a graph" by Jianxi Li, Wai Chee Shiu, and An Chang, Applied Mathematics Letters 23.3 (2010): 286-290, DOI:10.1016/j.aml.2009....
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2
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97
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Bound on the magnitude of the entries of the Laplacian pseudo-inverse
Let $L$ be the laplacian matrix of a connected graph $G$ with real positive weights and $N$ vertices, or that can be assumed to have binary weights for simplicity.My goal is to bound $\Vert L^+\Vert_{\...
- 21
0
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0
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128
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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
7
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4
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962
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Random sample of spanning trees
In a complete graph with $n$ vertices there are $n^{n-2}$ spanning trees.
I want to get a random sample of size $k$ from the set of all spanning trees.
The most basic and naive idea is to generate all ...
- 49
0
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0
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429
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What is the exact asymptotic bound on the following sum of polynomials?
I am trying to describe the asymptotic growth of the function $$f(n) = \sum_{k = 1}^{n-1} \frac{k^{2n - 4k - 3}(n^2 -2nk + 2k^2)}{(n-k)^{2n-4k-1}}$$ as $n \rightarrow \infty$. Plotting $f(n)$ for the ...
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97
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Relation of minimum spanning trees to the shortest Hamiltonian path problem
Spanning trees can be decomposed into a minimal set of maximal path graphs, whose vertices have degree two exactly if they also have degree two in the spanning tree; lets call these paths tree paths.
...
- 13.2k
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1
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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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1
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52
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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 (...
6
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5
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540
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Existence of connected set with large edge boundary
Let $\Gamma=(V,E)$ be a finite connected graph.
Pretty standard notation. Given a set $S\subset V$, write $\Gamma|_S$ for the restriction of $\Gamma$ to $S$, i.e., the subgraph $(S,\{\{v,w\}\in E: v,w\...
- 20.1k
1
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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 $...
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1
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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
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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
3
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1
answer
205
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Property of the spanning tree with minimal leaves
Let $G$ be a connected simple graph. For any spanning tree $T$ of $G$, let $l(T)$ be the number of leaves of the graph $T$. Consider $\ell=\min_Tl(T)$, can I find a spanning tree $T$ with $l(T)=\ell$, ...
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2
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91
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Blind construction of planar graph with additive spanning tree count
Suppose we have two planar graphs $G_1$ and $G_2$ with number of spanning tree count $P_1$ and $P_2$ respectively then there is an easy construction which gives a planar graph with spanning tree count ...
- 13.9k
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1
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158
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Heuristics for lightweighted "cubic" spanning trees
I have the problem of calculating a good approximation of the minimimum-weight spanning tree with vertex-degrees in $\lbrace 1,3\rbrace$ of a complete symmetric graph, without parallel edges or self-...
- 13.2k
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....
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1
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0
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27
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Optimality of trees generated via edge exchanges
let MST be the minimum spanning tree of a weighted finite graph; what can be said about the weight-optimality of the trees generated from the MST by sequentially exchanging a tree edge with a non-tree ...
- 13.2k
0
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0
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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
1
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0
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242
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How to estimate the probability of an edge appearing in the minimum spanning tree of a graph?
I've been running into this problem recently and I've been stuck on it for a while.
I have a set of vertices $G$ that form a complete graph. From this I need to sample $k$ vertices (which would also ...
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4
votes
1
answer
394
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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 ...
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1
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0
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240
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Relation between the number of spanning trees and the chromatic number of a graph
The number of spanning trees $\tau(G)$ of a simple graph $G$ is seen to satisfy the deletion-contraction recurrence:
$$\tau(G)=\tau(G-e)+\tau(G.e),$$
where $e$ is an edge of the graph $G$ and $G-e$ ...
- 2,089
2
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0
answers
162
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Calculating Minimum Spanning Trees in Very Big Graphs
I need to determine Minimum Spanning Trees (MST) of very big complete graphs, whose edgeweights can be calculated from data that is associated with the vertices.
In the planar euclidean case, for ...
- 13.2k
7
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1
answer
465
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Counting spanning trees of a planar graph
I know through Kirchoff's Theorem, one can calculate the number of spanning trees via the determinant of a Laplacian. This has complexity $O(N^{2.373}$). I was wondering if anyone was aware of a ...
- 3,499
7
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0
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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
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1
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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 $e, f$
$$\...
- 2,175
5
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1
answer
445
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Minimum euclidean spanning tree in n dimensional space
I need to compute the minimum euclidean spanning tree in $R^d$ and do it with some algorithm that can do it with complexity near to $\Omega(nlogn)$ where $n$ is the size of the point set.
Right now I'...
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2
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1
answer
124
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Hu-Gomory trees and Optimum Communication tree
It is known that that can be several trees in a graph that follow the conditions of "Cut-tree" (also called Hu-Gomory tree).
For example (https://stackoverflow.com/questions/25297470/igraphs-gomory-...
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5
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2
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460
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Spanning tree minimizing $F_T = \sum_{i = 1}^{|V| - 1|} (w(e_i) - P_T)^2$
Let $G = \langle V, E \rangle$ be an undirected, connected and weighted multigraph, with the weights given by a function $w: E \rightarrow N$. Consider any spanning tree $T$. Denote the edges of $T$ ...
- 161
10
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2
answers
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History of deletion-contraction formula
The following is known as deletion-contraction formula:
Assume $\Gamma$ is a connectted graph with edge $\rho$ then
$$t(\Gamma)=t(\Gamma\backslash\rho)+t(\Gamma/\rho),$$
where $\Gamma\backslash\...
- 45k
4
votes
0
answers
173
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average probability for an edge be a in random spanning tree of a weighted graph
any cofactor of a Laplacian of a weighted graph will give the sum of all weighted spanning trees, lets denote it by $A$. The same can be calculated for spanning trees which avoid certain edge $e$, ...
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6
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2
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917
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Create a graph with a specified number of spanning trees
I read that one of the current challenging problems in mathematics is constructing a minimal graph with a specified number of spanning trees (say, $k$).
However, is there a quick way to create some ...
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3
votes
2
answers
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 ...
- 173
0
votes
0
answers
165
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Expected length of minimum spanning trees
For a simple, finite, connected and complete graph $K_n = (V(K_n), E(K_n))$ with vertex set $V(K_n)$ and edge set $E(K_n)$, we assign a non-negative independent and identical distributed random weight ...
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0
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320
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Gromov-Hausdorff distance measure between minimum spanning trees
I am trying to compare minimum spanning trees through time. I have two questions:
1-Is it possible to measure the similarity between two minimum spanning trees with Gromov-Hausdorff distance measure ...
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3
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0
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265
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Characterizing graphs with $k$ edge-disjoint minimum diameter spanning trees
Henneberg [1] and Laman [2] characterized graphs which have, after adding any edge, 2 edge-disjoint spanning trees. This was generalized to $k$ edge-disjoint spanning trees by Frank and Szegõ [3]. ...
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5
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1
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Minimum Spanning Tree of Graph with Unknown Weights
I have a fully connected graph $G=(V,E)$ with $n$ vertices. The edge weights $w(e)$ with $e\in E$ are non-negative and form a metric space (e.g. Hamming distance), thus for vertices $v,u,y \in V$, we ...
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4
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Graphs in which every spanning tree is an independency tree
It follows from this question
and the corresponding answers, that the complete graphs and the cycles are precisely the graphs
$G$ having the property that, for every spanning tree $T$ of $G$, the ...
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6
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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 ...
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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\...
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1
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230
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Divisibility Relation for Spanning Trees of a Graph
Let $A = \big[{1\ 1\atop 1\ 0}\big]$, and let $G_n$ be the graph whose adjacency matrix is
$A^{\otimes n}$. Also let $\kappa(G)$ denote the number of spanning trees of $G$. From a significant amount ...
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1
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1
answer
1k
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Minimum spanning subgraph with at least one incoming and one outgoing edge
Given a single-component, directed acyclic graph with one source (vertex with only outgoing edges) and one sink (vertex with only incoming edges), I'd like to find a minimum spanning subgraph which ...
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3
votes
1
answer
2k
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How random are random spanning trees?
Suppose you take a $G(n,p)$ random graph for a fixed probability $p$ and find a spanning tree using Kruskal's algorithm. If you now repeat this process indefinitely, will every tree on $n$ vertices ...
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19
votes
4
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1k
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Minimal graphs with a prescribed number of spanning trees
As it's long ago since Erdős died and MathOverflow is the second best alternative to him (for discussing personal problems), I'd like to start a fruitful discussion about the following problem that I ...
- 3,463
6
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2
answers
1k
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Minimum spanning tree of a weighted graph
I have a connected graph $G=(V,E)$ in $n$ vertices. The edge weights are non-negative and form a metric space, thus for vertices $u,v,w \in V$ , such that $(u,v), (v,w), (w,u)\in E$ we have $r(u,w) \...
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1
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1
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3D Delaunay Triangulation -> Euclidean Minimum Spanning Tree
I read that the Euclidean Minimum Spanning Tree (EMST) of a set of points is a subgraph of any Delaunay triangulation. Apparently the easiest/fastest way to obtain the EMST is to find the Deluanay ...
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