All Questions
21 questions
2
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0
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48
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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\...
- 13.9k
2
votes
1
answer
105
views
"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 ...
- 48.8k
0
votes
0
answers
429
views
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 ...
0
votes
0
answers
97
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.
...
- 13.2k
1
vote
0
answers
52
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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 (...
6
votes
5
answers
540
views
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\...
- 20.2k
1
vote
1
answer
171
views
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 $...
- 1,295
1
vote
0
answers
378
views
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^{...
- 1,295
2
votes
1
answer
254
views
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 ...
- 1,295
3
votes
1
answer
205
views
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 $l(T)=\ell$, ...
- 622
2
votes
0
answers
91
views
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 ...
- 13.9k
1
vote
0
answers
240
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 $G-e$ ...
- 2,089
7
votes
1
answer
465
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 ...
- 3,499
7
votes
0
answers
171
views
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 ...
- 1,462
5
votes
1
answer
281
views
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 $e, f$
$$\...
- 2,175
10
votes
2
answers
1k
views
History of deletion-contraction formula
The following is known as deletion-contraction formula:
Assume $\Gamma$ is a connectted graph with edge $\rho$ then
$$t(\Gamma)=t(\Gamma\backslash\rho)+t(\Gamma/\rho),$$
where $\Gamma\backslash\...
- 45k
6
votes
2
answers
917
views
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 ...
- 171
3
votes
2
answers
287
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 ...
- 173
4
votes
1
answer
230
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 ...
- 43
19
votes
4
answers
1k
views
Minimal graphs with a prescribed number of spanning trees
As it's long ago since Erdős died and MathOverflow is the second best alternative to him (for discussing personal problems), I'd like to start a fruitful discussion about the following problem that I ...
- 3,463
6
votes
2
answers
1k
views
Minimum spanning tree of a weighted graph
I have a connected graph $G=(V,E)$ in $n$ vertices. The edge weights are non-negative and form a metric space, thus for vertices $u,v,w \in V$ , such that $(u,v), (v,w), (w,u)\in E$ we have $r(u,w) \...
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