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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.
2
votes
2
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437
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On Knot Equivalence problem statement
How is the knot equivalence problem represented?
By this I mean I am looking for an analogy that compares with graph equivalence. For graph equivalence, we have two graphs $G_1$ and $G_2$ with adjace …
- 2,934
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 …
CommunityBot
- 1
2
votes
0
answers
45
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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
3
votes
0
answers
98
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Number of planar bipartite graphs
How many planar bipartite graphs are there with $m$ vertices of one color and $n$ vertices of the other color?
How many non-isomorphic classes exist?
- 13.9k
2
votes
0
answers
102
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Counting number of perfect matchings
Counting perfect matchings in bipartite graphs is $\# P$ complete. Let $G(V,E)$ be a graph known to have $d$ number of perfect matchings. Bipartite it the obvious way by adding $E$ vertices with one v …
- 63.9k
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 $G$?
- 4,713
2
votes
0
answers
91
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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
7
votes
2
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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 …
- 32.1k
2
votes
0
answers
60
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Sum of number of perfect matchings and a constant constuction
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
0
answers
244
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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
3
votes
1
answer
158
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Probability permutation in turned to cycle
Let $M$ be a $0/1$ square matrix having one $1$ per row and column (permutation matrix).
If you permute the columns and rows independently what is the probability resulting permutation matrix is a com …
- 128k
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 …
- 761
3
votes
1
answer
95
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On cospectral graphs
Is there examples of non-isomorphic cospectral graphs having
Non-isomorphic automorphism groups?
Isomorphic automorphism groups?
- 14k
1
vote
1
answer
303
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Unique bipartite perfect matchings and cycles?
Given a graph $G$ which is bipartite and balanced and has unique perfect matching let $G^{e}$ be $G$ without edge $e$. Let $G\cup G_{\pi,\pi'}$ be union of $G$ and $G_{\pi,\pi'}$ where $G_{\pi,\pi'}$ …
- 1,042
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0
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112
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Is there a bipartite graph whose determinant corresponds to number of perfect matchings?
Let $M\in\{0,1\}^{n\times n}$ be a square integer matrix. If we consider $M$ as biadjacency of a balanced bipartite graph on $2n$ vertices having $n$ vertices of color $1$ and $n$ vertices of color $2 …
- 13.9k