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Results tagged with graph-theory
Search options questions only
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user 10035
Questions about the branch of combinatorics called graph theory (not to be used for questions concerning the graph of a function). This tag can be further specialized via using it in combination with more specialized tags such as extremal-graph-theory, spectral-graph-theory, algebraic-graph-theory, topological-graph-theory, random-graphs, graph-colorings and several others.
18
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8
answers
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Concepts in topology successfully transferred to graph theory and combinatorics with non-tri...
What are some of the difficult concepts in topology that have been transferred to graph theory and combinatorics where a certain new application has been found.
A good example is Lovász's proof of Kn …
14
votes
2
answers
1k
views
Name for algebra and its tensor products
Is there a name for the algebra (and its tensor products) given by generators $M_{j}$, $j \in \mathbb{Z}_{n}$ under the conditions $$M_{j} = (1 - M_{j-1})(1-M_{j+1})$$ where $j=1\implies j-1=n$ and $ …
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14
votes
3
answers
1k
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A question on representation of graphs
Take a complete graph $K_n$. You want to assign a vectors from $\Bbb F_2^d$ to every edge such that sum of vectors in every simple cycle does not sum to $0$ vector. The question is what is minimum $d$ …
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9
votes
1
answer
1k
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Lovasz theta function and independence number of product of simple odd-cycles
Lovasz theta function $\vartheta(G)$ of a graph $G$ provides an upper bound for the independence number of a graph, $\alpha(G)$ and $\Theta(G) = \lim_{k\rightarrow \infty}\sqrt[k]{\alpha(G^{k})}$. Tha …
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8
votes
0
answers
244
views
Sum of perfect matching construction
Suppose we have two bipartite graphs $G_1$ and $G_2$ with perfect matching count $P_1$ and $P_2$ respectively then their disjoint union gives a bipartite graph with perfect matching $P_1P_2$.
Is ther …
- 13.9k
8
votes
2
answers
741
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Maximum number of perfect matchings in a planar graph?
What is the maximum number of perfect matchings a planar $k$-partite $|V|$ number of vertices simple graph can have where $k=2,3,4$ ($k>4$ is impossible for a planar graph)?
Since number of matchi …
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7
votes
1
answer
950
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Graph to Bipartite conversion preserving number of perfect matchings
Given a graph $G$ on $n$ vertices is there a technique to convert to a balanced bipartite graph $B$ with $O(n^c)$ vertices at some fixed $0<c$ in $O(n^{c'})$ time at some fixed $0<c'$ such that the n …
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7
votes
0
answers
347
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Missing count in number of perfect matchings
Let $f(G)$ give number of perfect matchings of a graph $G$.
Denote $\mathcal N_{2n}=\{0,1,2,\dots,n!-1,n!\}$.
Denote collection of all $2n$ vertex balanced bipartite graph to be $\mathcal G_{2n}$.
…
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7
votes
2
answers
814
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Graph minor check
Are there any good algebraic/algorithmic tools available to check if a given graph $H$ is a minor of $G$ from the adjacency matrix of $G$?
- 13.9k
7
votes
2
answers
478
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Are all numbers from $1$ to $n!$ the number of perfect matchings of some bipartite graph?
Let $f(G)$ give the number of perfect matchings of a graph $G$.
Consider set $\mathcal N_{2n}=\{0,1,2,\dots,n!-1,n!\}$.
Consider collection of all $2n$ vertex balanced bipartite graph to be $\mathca …
- 13.9k
6
votes
2
answers
359
views
Who first used/gave a coordinate representation of a graph?
In his proof of the Shannon capacity of a graph, Lovasz utilizes a coordinate representation of the pentagon (namely an orthonormal representation). Who first utilized a coordinate representation for …
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6
votes
3
answers
422
views
Repository of graph classes that are tough to test non-isomorphic pairs from isomorphic pairs
(1) Which graph classes are extremely tough to test for graph non-isomorphic pairs from isomorphic pairs?
(2) Is there a repository of adjacencies from such classes?
- 13.9k
5
votes
2
answers
837
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Database of adjacency matrices on cospectral non-isomorphic graph pairs
Is there a repository of cospectral non-isomorphic graphs available somewhere?
I am looking for list of $0/1$ adjacency matrix pairs that can be input data in tools such as MATLAB.
- 13.9k
5
votes
3
answers
868
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On number of perfect matchings
Consider $2n$ vertex balanced bipartite graph.
If total number of edges is $n^2$ then we have $n!$ perfect matchings.
Fix $c\in(0,\frac12)$ and consider collection of $2n$ vertex balanced bipartite …
- 13.9k
5
votes
1
answer
611
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Lovász theta and circulant graphs
Let $\theta(G)$ denote the Shannon zero error capacity of graph $G$ and $\vartheta(G)$ be the Lovász upper bound for $\theta(G)$.
Let $C_{2n+1}$ denote the cycle graph with $2n+1$ nodes.
We know the f …
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