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Results tagged with graph-theory
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user 1492
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.
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What are the efficient algorithms to compute Hamiltonian paths on Cayley graphs of finite gr...
I do not have a definitive answer, but some thoughts that are too long to go in a comment - I left the question open for a few days in case someone else knew more.
It seems to be the case that all ver …
- 12.7k
4
votes
Accepted
Is there a ternary Cayley graph on 27 vertices that is a non-complete core?
There are two such graphs on $27$ vertices, one with degree $12$ and one with degree $14$.
To get the degree $14$ example, just add $j+k$ to the connection set of your degree $12$ example.
Both of the …
- 12.7k
1
vote
Accepted
Bounds on the number of proper 3-colorings of cubic graphs
I don't have a complete answer here, but some things that might help.
First I am just going to assume that we want to maximise the value $P_G(3)$ where $P_G$ is the chromatic polynomial of $G$.
There …
- 12.7k
1
vote
Rank of adjacency matrix of a graph on a sphere all of whose faces have four vertices
This is not exactly an answer, but too long for a comment.
In general, the rank is not a very useful graph parameter, mostly you expect graphs to have full rank unless they cannot.
However it doesn't …
- 12.7k
8
votes
A new basis for chromatic polynomials
In case anyone stumbles across this question, the statement (that all the coefficients in this basis are non-negative) is not true.
The complete bipartite graph $K_{7,7}$ has the following coefficient …
- 12.7k
4
votes
Multiplicity of the smallest non-zero Laplacian eigenvalue for tree graphs
All the Laplacian eigenvalues of the star graph $K_{1,n}$ other than $n+1$ and $0$ are equal to $1$.
Computer experimentation reveals a modest number of additional examples. One pattern that may gener …
- 12.7k
5
votes
Accepted
When do the nonzero eigenvalues of a directed graph Laplacian have the same absolute value?
This is false.
Run this code in SageMath; you can do this at sagecell.sagemath.org if you do not have SageMath already installed on your own computer.
h = DiGraph('DKCYW?')
print(h.laplacian_matrix(). …
- 12.7k
10
votes
Accepted
Generating 21-vertex 4-regular plane graphs with 8 faces of degree 3 and 15 faces of degree 4
Brinkmann and McKay's program plantri can generate planar quadrangulations, which are planar graphs with all faces of size 4.
If you generate these on 23 vertices and then filter to keep only those of …
- 12.7k
1
vote
Random sample of spanning trees
There is an algorithm for unranking and ranking spanning trees due to Colbourn, Day and Nel which is described in https://www.sciencedirect.com/science/article/abs/pii/0196677489900163
You can compute …
- 12.7k
10
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Is there a bipartite graph with $\sqrt{2}$ as an eigenvalue with high multiplicity, specific...
This is a limited partial answer to the question ruling out the case where the graph is regular and has four distinct eigenvalues, so the spectrum is $\{k, (\sqrt{2})^a, (-\sqrt{2})^a, -k\}$.
van Dam' …
- 1,446
3
votes
Accepted
How can one construct a class of $k$-connected $k$-regular bipartite graphs with the girth o...
I'll answer for $k=5$ and leave the remaining cases for you to generalise.
For $n \geqslant 5$, let $X_n$ denote the Cayley graph $\mathrm{Cay}(\mathbb{Z}_{2n},\{\pm 1, \pm 3, n\})$.
Claim
The graph $ …
- 12.7k
4
votes
Accepted
Edge coloring of a graph on alternating groups
This graph is 4-colourable.
Here is some SageMath code that constructs the graph, forms its line graph and then presents a 4-colouring of the graph that you can check.
I found the colouring using the …
- 12.7k
1
vote
Accepted
Eigenvalues of directed graph with one outward edge for each vertex
Here is an alternative (more combinatorial) proof to the one linked to in my comment.
Suppose that the digraph $D$ has a vertex of in-degree zero, which we may assume is vertex $1$. Then letting $\var …
- 12.7k
21
votes
Accepted
Is this graph Hamiltonian?
Here is a 9-vertex graph with degree sequence 4 4 4 4 4 3 3 3 3 that does not have a Hamilton cycle (because it is bipartite on an odd number of vertices).
Edit: Here is one line of SageMath showing …
- 12.7k
5
votes
Accepted
3-coloring the alternating group graph
So I believe that $\chi(A_7) > 3$, but this relies on some computations that require verification.
But I will start by giving some SageMath code that computes the graph, finds an independent set of si …
- 12.9k