The spanning-tree tag has no usage guidance.

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### Similarity metric for labelled weighted graph/minimum spanning tree

I'm looking for a metric to measure similarity of minimum spanning trees of labelled weighted graphs. Each entity to compare has the same nodes (number of nodes and labels identical) but (unlabelled) ...

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281 views

### 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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164 views

### 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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237 views

### 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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148 views

### 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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234 views

### 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 ...

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### On some special spanning trees of grid graphs

I would like to know if there are existing results on the following objects:
spanning trees of a grid graph, with no corridor
where a corridor is a vertex having exactly two neighbors, on ...

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457 views

### 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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952 views

### 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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**1**answer

334 views

### 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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235 views

### Matrix-tree for matrices with constant diagonal

I've got a symmetric matrix $A$ whose entries are in $\{0,-1,1\}$, with the diagonal entries all equal to $1$. I'm interested in finding a combinatorial description of the entries of the inverse of ...

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136 views

### rainbow spanning tree

In graph G, every edge has a color. Rainbow spanning tree is a spanning tree where all edges have different colors.
I want a polynomial algorithm to find such tree if exists any
Anyone can help?

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403 views

### 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 ...

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504 views

### Minimize diameter of a tree

Hi! I have an acyclic undirected unweighted connected graph (a tree :) ), and I have to disconnect an edge and create a new one to minimize the diameter.
For now, I do a bfs on a arbitrary node, find ...

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184 views

### 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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491 views

### 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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607 views

### 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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### 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 ...