# Questions tagged [spanning-tree]

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### Online algorithms for MSTs from time series

Is it possible to construct the MST (minimum-weight spanning tree) for an potentially infinite sequence of points $\lbrace (i,Y[i]): i\in\mathbb{N}_0,\,c_{\text{min}} \le Y[i]\le c_{\text{max}}\rbrace$...
• 12.8k
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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
846 views

### 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
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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 ...
86 views

### 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. ...
• 12.8k
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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) ...
• 47
1 vote
171 views

### Relation of MSTs in the Euclidean plane to Delaunay triangulations

It is known that the Minimum Spanning Tree (MST) of a finite set of points in the Euclidean plane is contained in the point set's Delaunay triangulation, but is that all that can be said about their ...
• 12.8k
1 vote
48 views

### 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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• 1,243
1 vote
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• 12.8k
1 vote
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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 ...
• 12.8k
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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 ...
• 19.3k
1 vote
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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 ...
• 11
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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 ...
1 vote
217 views

### 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 $G-e$ ...
• 2,079
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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 ...
• 12.8k
457 views

### 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 ...
• 3,469
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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 ...
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• 23.1k
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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 ...
• 3,949
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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
161 views

### 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 ...
• 1
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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 ...
• 1
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### Euclidean Minimum Spanning Trees Restricted to One Vertex Per Grid Cell

Given an $n \times n$ grid with unit grid cells, and one point from the interior of each cell, what is are best possible lower and upper bounds for lengths of minimum spanning trees? The lower bound ...
### 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]. ...