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Results tagged with graph-theory
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user 11260
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.
4
votes
Multiplicity of the smallest non-zero Laplacian eigenvalue for tree graphs
The multiplicity of Laplacian eigenvalues of tree graphs is studied in arXiv:1907.11482. If $\Delta$ is the maximal degree of the graph, and $\Delta\geq 2$, then the multiplicity $m_2$ of $\lambda_2$ …
- 188k
1
vote
Is there a version of Weyl's law for graph Laplacians?
For $N_\lambda$ denoting the number of eigenvalues less than $\lambda$, Weyl's law gives the asymptotics of $N_\lambda$ as $\lambda$ tends to infinity. The usual approach to establish this asymptotics …
- 188k
9
votes
Accepted
Has Plummer's open problem on the cyclic connectivity of planar graphs been solved?
Yes, it has been solved.
In 1989 Borodin proved that the maximum cyclic edge connectivity of a 5-connected planar graph is at most 11, improving on Plummer's upper bound of 13. The 11 bound is tight [ …
- 188k
4
votes
Accepted
A reliable reference for the statement every $k$-tree is uniquely $(k + 1)$-colorable
I presume there are many "reliable references", here is one: Well-covered Graphs, Unique Colorability, and Covering Range, it's a 6-line proof at the bottom of page 4.
- 188k
3
votes
Accepted
A reference for Wagner's Theorem
A text book reference is Graph Decompositions: A Study in Infinite Graph Theory by Reinhard Diestel (1990), see chapter 6.
- 188k
3
votes
Accepted
First known proof of the $2 \cdot n-2$ Theorem for the planar generalization of the Nine dot...
This might satisy the "formally correct" criterion: Unicursal polygonal paths and other graphs on point lattices by Golomb and Selfridge (1970). It generalizes and extends the 1955 proof by Selfridge, …
- 188k
6
votes
Accepted
The origin of a planar graph theorem of Steinitz and Rademacher
According to Frank Lutz's article it's in paragraph 46 of the Steinitz-Rademacher book: "every triangulated 2-sphere can be reduced to the boundary of the tetrahedron by a sequence of edge contraction …
- 188k
3
votes
About a result by P. Erdős and H. Sachs on graph with large girth
I OCR'd the text from the abstract (the pdf contains poor quality text, OCR'ing it yourself gives better results), pasted the OCR'ed text into DeepL, and received the following, without any edits from …
- 188k
10
votes
How to effectively search Internet for graphs not for function graphs?
Use Google Scholar instead of plain Google; I simply entered graphs and pretty much all the items returned by Google Scholar refer to graphs in the mathematical context.
It also suggests helpful speci …
- 188k
2
votes
Accepted
Qualitative values between two electrons in an atom or how to interpret these values?
The key physics that governs the ionisation energies is shell formation; electrons in the same atomic shell have similar ionisation energies; the number of electrons in the $n$-th shell is $2n^2$, so …
- 188k
2
votes
Accepted
Who introduced the concept of beyond planar graphs?
A pre-Dagstuhl reference is
Graph drawing beyond planarity: some results and open problems.
G Liotta,ICTCS 14, 3-8 (2014).
The “beyond planarity” research area could be briefly described as the
(pote …
- 188k
6
votes
Is there any fast implementation of four color theorem in Python?
Here is a greedy algorithm by Febi Mudiyantoto solve the four-color problem in Python.
And here is another Python algorithm that also uses Sage.
If you wish to rely on a program with a more formal (re …
- 4,713
3
votes
What is the definition of brick product of graphs?
On a class of Hamiltonian laceable 3-regular graphs, contains the definition of the brick product of even cycles, with figures showing examples.
Definition. Let $m$, $n$ and $r$ be a positive integer …
- 1,446
2
votes
Accepted
Is there a monograph or review of Hamiltonian cycles of graphs (or long cycles of graphs)？
Q: Is there a monograph (or review) of Hamiltonian cycles of graphs (or long cycles of graphs)？
One possible answer (from a specific perspective) is
Hamiltonian Cycle Problem and Markov Chains (2012)
…
- 188k
4
votes
Accepted
Why is the spectrum of Erdős–Renyi random graph approximately symmetric?
The Erdős–Renyi random matrix $G$ is a rank-one perturbation of a matrix $H$ with zero mean,
$$G=H+pv^\top v,\;\;v=(1,1,\ldots 1)^\top.$$
The spectrum of $H$ is symmetric, while the eigenvalues $\lamb …
- 188k