# Questions tagged [spanning-tree]

The spanning-tree tag has no usage guidance.

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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\...

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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 ...

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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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### 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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### 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_{\...

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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 $\lbrace (i,Y[i]): i\in\mathbb{N}_0,\,c_{\text{min}} \le Y[i]\le c_{\text{max}}\rbrace$...

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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 ...

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4
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846
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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 ...

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419
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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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### 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.
...

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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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### 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 ...

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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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### 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\...

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### Class numbers of functions fields and spanning trees

In Discrete groups, expanding graphs, and invariant measures (in the notes at the end of Chapter 7), Lubotzky mentions that certain estimates for the number of spanning trees $\kappa(G)$ of a $k$-...

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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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279
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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^{...

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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 ...

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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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### 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 ...

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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-...

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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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### Calculating number of vertex-pairs with separate common ancestor

Given a tree-graph with one of the vertices designated as the root, what is the complexity of calculating the number of vertex-pairs $\lbrace u,v \rbrace$ of which $v$ is not nearer to the root than $...

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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 ...

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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 ...

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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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357
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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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### 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 $G-e$ ...

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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 ...

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457
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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 ...

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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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### 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 $e, f$
$$\...

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### Does the zeta regularized Laplacian determinant measure the volume of some parameter space? How many "spanning trees" on a manifold?

Let $(M,g)$ be a Riemannian manifold, with Laplacian $\Delta$. If $\lambda_i$ are the nonzero eigenvalues of $\Delta$, we can define the zeta function $\zeta(s) = \Sigma \lambda_i^{-s}$. By analytic ...

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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 $n$...

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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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### How to find a minimal rooted tree with maximial sum vertex weights?

I have an undirected grid graph, nxm, where each vertex has a value (positive or negative) and I need to find the tree rooted at vertex (1,1) that maximizes the sum of these values with a minimal ...

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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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### 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 $T$ ...

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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\...

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1
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### Spanning tree with sufficient transformation [closed]

How can I give a set of transitions sufficient to transform any spanning tree into any another spanning tree in a finite number of steps via spanning trees? I was wondering if someone help me.Thanks.

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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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### 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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### Even parking functions and spanning trees of complete bipartite graphs

Set $\mathbb{N} := \{0,1,2,\ldots\}$. A parking function of length $n$ is a sequence $(\alpha_1,\ldots,\alpha_n) \in \mathbb{N}^n$ whose weakly increasing rearrangement $\alpha_{i_1} \leq \alpha_{i_2} ...

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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 ...

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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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### 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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### 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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### 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 ...

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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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### Maximum number of hyperedges on a hypergraph without a spanning tree

Although every connected graph has a spanning tree, the same is not true for hypergraphs: consider the hypergraph on 4 vertices with all possible edges of size 3. You need to pick at least two edges ...