4
votes
2answers
168 views

Are the following Cayley digraphs Hamiltonian?

Consider the Cayley graphs $A'G_n$ on the alternating group $A_n$ with generating set $S = \{(1i2) : 3 \leq i \leq n\}$, for $n \geq 4$. See the following page on Alternating Group Graphs for ...
1
vote
1answer
116 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)?
1
vote
0answers
126 views

A connection between nonplanar complete graphs and the alternating group?

I didn't get any response on MSE so I though I'd give this a try here (my question on MSE). I went to an undergrad's senior honors thesis presentation a while ago. She was discussing crossing numbers ...
7
votes
3answers
494 views

Is there a Cayley graph of a non-abelian group that is not isomorphic to any Cayley graph of any abelian group?

It's the first question I post here :) I'm sorry if the question is too specific or if it's somehow repeating others. In other words, my question is the following. Consider a Cayley graph $\Gamma$ of ...
2
votes
2answers
304 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 ...
13
votes
0answers
220 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 ...
2
votes
2answers
194 views

Mclaughlin Graph

how can i construct a strongly regular graph with parameter $(275,112,30,56)$(Mclaughlin Graph), (105,32,4,12)? I need adjacency matrix of them? I know they are unique.
11
votes
3answers
755 views

“Antipodal” maps on regular graphs?

This question is related to Realizing the diameter of a finite regular graph Let $X=(V,E)$ be a finite, connected, regular graph of diameter $D$. Assume that, for every vertex $x\in V$, there exists ...
15
votes
3answers
1k views

What is this subgroup of $\mathfrak S_{12}$ ?

On some occasion I was gifted a calendar. It displays a math quizz every day of the year. Not really exciting in general, but at least one of them let me raise a group-theoretic question. The quizz: ...
7
votes
2answers
701 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 ...
23
votes
9answers
2k views

How many groups of size at most n are there? What is the asymptotic growth rate? And what of rings, fields, graphs, partial orders, etc.?

Question. How many (isomorphism types of) finite groups of size at most n are there? What is the asymptotic growth rate? And the same question for rings, fields, graphs, partial orders, etc. ...