All Questions
Tagged with symmetric-groups graph-theory
9 questions
3
votes
1
answer
149
views
Drunken X-mas polynomials for graphs
Given a finite connected graph $\Gamma$ with vertices $\lbrace 1,\ldots,N\rbrace$,
we can consider the polynomial
$$\sum_{\pi\in\mathcal S_N}x^{\sum_{j=1}^Nd_\Gamma(j,\pi(j))}$$
where $\mathcal S_N$ ...
- 17.9k
7
votes
2
answers
421
views
3-coloring the alternating group graph
Consider the alternating group graph, here defined as a Cayley graph on the alternating group $A_n$ using the generating set $\{(1,2,3),(1,2,4),\dotsc,(1,2,n),(1,n,2),\dotsc,(1,4,2),(1,3,2)\}$. Note ...
- 2,089
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
3
votes
0
answers
56
views
Groups that can occur as graph automorphisms of a fixed size graph
From theorem $4$ and corollary $1$ in this book we have that graph isomorphism has to do with automorphism group of a graph. We also know every group is the automorphism group of a graph by Frucht's ...
- 13.9k
10
votes
1
answer
269
views
Edge-transitive Cayley graphs of $S_n$
I came across the following question which I haven't seen before:
Question. Fix $k\ge 3$. For infinitely many $n$, does there exists a generating set $\langle R_n \rangle = S_n$, $|R_n|=k$, such ...
- 17k
3
votes
0
answers
164
views
Generating sets of the symmetric group that yield isomorphic Cayley graphs
Let $S$ and $S'$ be subsets of size $k$ of $\mathfrak{S}_n$.
Are there any necessary or sufficient conditions to determine whether or not $S$ and $S'$ yield isomorphic Cayley graphs?
Assuming we ...
- 1,164
6
votes
3
answers
434
views
Repository of graph classes that are tough to test non-isomorphic pairs from isomorphic pairs
(1) Which graph classes are extremely tough to test for graph non-isomorphic pairs from isomorphic pairs?
(2) Is there a repository of adjacencies from such classes?
- 13.9k
9
votes
1
answer
356
views
Diameter of the modified bubble-sort graph
The modified bubble-sort graph is the Cayley graph $Cay(S_n,S)$ of $S_n$ generated by $n$ cyclically adjacent transpositions. Thus $S = \{ (1,2),(2,3),\ldots,(n,1)\}$. I was wondering whether the ...
- 406
12
votes
4
answers
2k
views
Cyclic Permutations - but not what you think
This question is not about elements of $S_n$ that consist of a single $n$-cycle, though naturally it's related.
Instead, consider permutations modulo the action of $(123\ldots n)$. That is, we ...
- 367