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**5**

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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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### Can we make the weaker TPC tighter when we do the packing steiner trees?

The famous tree packing conjecture (TPC) posed by Gyarfas (see [1]) states:
$\textbf{Conjecture 1}$. Any set of $n ā 1$ trees $T_n, T_{nā1}, . . . , T_2$ such that $T_i$ has $i$ vertices pack into ...

**12**

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**4**answers

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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 ...

**5**

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**1**answer

265 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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### 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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**0**answers

51 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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**0**answers

180 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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**3**answers

180 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 ...

**4**

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**1**answer

166 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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**1**answer

265 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 ...

**3**

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**1**answer

283 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 ...