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Results tagged with graph-theory
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user 4400
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.
3
votes
Accepted
Representing repeated structure in graphs
I am not sure what exactly is meant by a repeated structure but at least some covering graphs should qualify. Covering graphs may be quite large, however they admit concise description via voltage gr …
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9
votes
1
answer
419
views
A graph on irrationals where p is adjacent to q if p^q or q^p is rational.
When I was in high school I learned about an elementary proof that there exist irrational numbers $p$ and $q$ such that $p^q$ is rational. Put $p = q = \sqrt{2}$; if $p^q$ is rational, we are done. Ot …
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1
vote
Properties of Graphs with an eigenvalue of -1 (adjacency matrix)?
Maybe you should ask which of the eigenvalues should have value -1. For instance when Patrick Fowler and I explored the middle eigenvalue $\lambda_n$ in the decreasing sequence of eigenvalues of a gr …
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1
vote
Why is edge-coloring less interesting than vertex-coloring?
I do not think that edge-coloring is necessarily less interesting than vertex-coloring. I work in neither but have used both in my research. I have a feeling that vertex-coloring may be perceived as m …
5
votes
Looking for cubic, bipartite graphs with girth at least six and no cycles of length 8.
Desargues graph, also known as the generalized Petersen graph $G(10,3)$ has girth 6 and also contains cycles of length 8. There exist three 10-cages, smallest cubic graphs of girth 10. They have 70 ve …
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12
votes
2
answers
973
views
Drawing 3-configurations of points and lines with straight lines
It is well-known that the black-and-white coloring of the Heawood graph on 14 vertices determines a combinatorial 3-configuration with 7 "points" and 7 "lines", known as Fano plane. Similarly, any cub …
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4
votes
Checking if two graphs have the same universal cover
If two graphs have a common cover, then they have a common universal cover. The "maximal" cover is therefore unique. In general, the reverse is not true. The "minimal" common cover may not be unique. …
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6
votes
1
answer
1k
views
Finding a cycle of fixed length in a bipartite graph
Is finding a cycle of fixed even length in a bipartite graph any easier than finding a cycle of fixed even length in a general graph? This question is related to the question on Finding a cycle of fix …
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6
votes
3
answers
1k
views
Is there a bipartite analog of graph theory?
I would like to compile a list of questions about graphs that have a non-trivial analogs for bipartite graphs.
Let me give the following examples:
Cycle vs Even cycle. Most questions about cycles i …
3
votes
0
answers
316
views
Drawing a combinatorial 3-configuration of points and lines with pseudolines
This question is related to the question of drawing a combinatorial 3-configuration of points and lines with straight lines. We only relax the condition and admit drawings with pseudolines. Let us cal …
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1
vote
Accepted
Are combinatorial configurations whose Levi graphs may be represented as covering graphs ove...
The fist part of your question has a negative answer, since both Fano plane (73) and Moebius-Kantor configuration (83) are cyclic configurations. Here it is shown that the cyclic covering graphs over …
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3
votes
Generalizations of Planar Graphs
A group having a planar Cayley graph is sometimes called planar. Finite planar groups are well understood. The situation with infinite planar groups and their Cayley graphs is much more complicated; i …
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21
votes
5
answers
2k
views
The middle eigenvalues of an undirected graph
Let $ \lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_{2n} $
be the collection of eigenvalues of an adjacency matrix of an undirected graph $G$ on $2n$ vertices. I am looking for any work or references …
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