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2
votes
0answers
79 views

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]. ...
2
votes
1answer
119 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 ...
11
votes
0answers
107 views

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 ...
5
votes
1answer
236 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 ...
12
votes
4answers
913 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 ...
6
votes
1answer
285 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 ...
3
votes
0answers
133 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 ...
3
votes
1answer
83 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?
2
votes
0answers
263 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 ...
0
votes
3answers
257 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
votes
1answer
174 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 ...
1
vote
1answer
336 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
votes
1answer
390 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 ...