All Questions
17 questions
17
votes
0
answers
512
views
Maximum automorphism group for a 3-connected cubic graph
The following arose as a side issue in a project on graph reconstruction.
Problem: Let $a(n)$ be the greatest order of the automorphism group of a 3-connected cubic graph with $n$ vertices. Find a ...
- 37.7k
12
votes
2
answers
1k
views
Graph automorphism group
Let $A_w$ denote such set of positive integer $n$ that: for any two permutations $\pi_0,\pi_1\in S_n$, if $\pi_1$ is not a power of $\pi_0$, then there exists a (labeled non oriented) graph $G$ of ...
- 909
10
votes
1
answer
906
views
Which finite groups are not the automorphism group of some rooted finite tree?
The question is as given in the title:
Which finite groups are not the automorphism group of some rooted finite tree?
A rephrasing could be: Is any finite group representable as the automorphism ...
- 1,306
8
votes
2
answers
1k
views
Are vertex and edge-transitive graphs determined by their spectrum?
A graph is called vertex and edge transitive if the automorphism group is transitive on both vertices and edges.
The spectrum of a graph is the collection (with multiplicities) of eigenvalues of the ...
- 16k
7
votes
1
answer
517
views
Paths in groups
Given a finite group $G$, write $K(G)$ for the complete digraph on the elements of $G$. Label the edge from $g$ to $h$ by element $g^{-1}h$.
Question: For what groups does there exist a Hamiltonian ...
- 17.6k
7
votes
1
answer
337
views
Lovasz's conjecture for dihedral Cayley graphs
Background:
A tantalizing conjecture of Lovasz is the following:
Let $G$ be a (finite) connected vertex-transitive graph. Then $G$ contains a Hamiltonian cycle or is one of $5$ counter-examples.
(...
- 3,499
5
votes
2
answers
567
views
Orbits of independent sets of the hypercube
How does one enumerate the distinct orbit classes of independent sets of the hypercube modulo symmetries of the hypercubes?
The counting of the number of independent sets in an $n$-dimensional ...
- 171
5
votes
1
answer
275
views
Diameter of Cayley graphs of finite simple groups
Babai, Kantor and Lubotzky proved in 1989 the following theorem (Sciencedirect link to article).
THEOREM 1.1. There is a constant $C$ such that every nonabelian finite simple group $G$ has a set $S$ ...
- 237
4
votes
2
answers
485
views
Transposition Cayley graphs are planar
Consider the Cayley graph $G$ with vertex set the elements of the symmetric group $S_n$ and generating set the set of minimal transposition generators of the group $S_n$, that is the set $S=\{(12),(13)...
- 2,089
4
votes
2
answers
1k
views
Automorphism group action leads to a "quotient graph"
Let $G$ be a simple (finite) graph. Consider the next natural equivalence relation $\sim$ on $V(G)$:
$u\sim v$ iff there exists and automorphism $\phi\in Aut(G)$, such that $\phi(u)=v$.
Define a new ...
- 747
4
votes
0
answers
95
views
Is the size of maximum matching in vertex transitive 3-uniform hyper-graph on $n$ vertices always $\Omega(n)$?
What is the best known lower bound on the size of the maximum matching in a vertex transitive $3$-uniform hyper-graph?
- 197
2
votes
0
answers
85
views
G graph connections for finite groups G
In my research, I have seen G graph connections usually when G is a Lie group and the graph is the fatgraph of a (punctured) surface. This is usually in a physics context. However, I am curious to ...
- 495
2
votes
0
answers
140
views
About the eigenvectors of a matrix related to a Cayley graph
In some papers about the cayley graphs of finite groups the behaviour of the eigenvalues and eigenvectors of $\phi$ were discussed when $\phi=\sum_{g\in G} \lambda_G(g)$ and $\lambda_G(g)$ is defined ...
- 21
1
vote
1
answer
155
views
Max order for which connected Cayley Graphs are known to be Hamiltonian
There is a well-known conjecture that all connected Cayley graphs are Hamiltonian.
For how large a value of n has the conjecture been verified (i.e., for all groups whose order is at most n)?
- 349
1
vote
1
answer
178
views
Does every connected vertex transitive graph on $n$ vertices (except for $C_n$) have minimum feedback vertex set of size $\Omega(n)$?
Feedback vertex set is a set of vertices whose removal leaves an acyclic graph.
It is known that every vertex transitive graph on $n$ vertices has minimum vertex cover of size $\Omega(n)$. It is also ...
- 197
1
vote
1
answer
80
views
What are the efficient algorithms to compute Hamiltonian paths on Cayley graphs of finite groups ? Can GAP do it?
The famous Lovasz conjecture predicts existence of the Hamiltonian path on Cayley graphs. In general finding such a path is NP-complete problem, but there are many heuristic algorithms.
Question 1: ...
- 24.9k
-1
votes
1
answer
215
views
Perfect Cayley graphs for abelian groups have $\frac{n}{\omega}$ disjoint maximal cliques
Let $G$ be a perfect/ weakly perfect Cayley graph on an abelian group with respect to a symmetric generating set. In addition let the clique number be $\omega$ which divides the order of graph $n$. ...
- 2,089