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Let $F_q$ be a finite field of characteristic 2. Let $x^2 + Sx +P \in F_q[x]$ be an irreducible polynomial over $F_q$, and let $g$ be one of its roots in $F_{q^2}$. Define a map $M: ... 1answer 354 views Paths in groups Given a finite group$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$. Question: For what groups does there exist a Hamiltonian ... 2answers 173 views Are the following Cayley digraphs Hamiltonian? Consider the Cayley graphs$A'G_n$on the alternating group$A_n$with generating set$S = \{(1i2) : 3 \leq i \leq n\}$, for$n \geq 4$. See the following page on Alternating Group Graphs for ... 1answer 121 views Max order for which connected Cayley Graphs are known to be Hamiltonian There is a well-known conjecture that all connected Cayley graphs are Hamiltonian. For how large a value of n has the conjecture been verified (i.e., for all groups whose order is at most n)? 0answers 132 views A connection between nonplanar complete graphs and the alternating group? I didn't get any response on MSE so I though I'd give this a try here (my question on MSE). I went to an undergrad's senior honors thesis presentation a while ago. She was discussing crossing numbers ... 3answers 513 views Is there a Cayley graph of a non-abelian group that is not isomorphic to any Cayley graph of any abelian group? It's the first question I post here :) I'm sorry if the question is too specific or if it's somehow repeating others. In other words, my question is the following. Consider a Cayley graph$\Gamma$of ... 2answers 331 views Automorphism group action leads to a “quotient graph” Let$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$. Define a new ... 0answers 248 views Maximum automorphism group for a 3-connected cubic graph The following arose as a side issue in a project on graph reconstruction. Problem: Let$a(n)$be the greatest order of the automorphism group of a 3-connected cubic graph with$n$vertices. Find a ... 2answers 197 views Mclaughlin Graph how can i construct a strongly regular graph with parameter$(275,112,30,56)$(Mclaughlin Graph), (105,32,4,12)? I need adjacency matrix of them? I know they are unique. 3answers 767 views “Antipodal” maps on regular graphs? This question is related to Realizing the diameter of a finite regular graph Let$X=(V,E)$be a finite, connected, regular graph of diameter$D$. Assume that, for every vertex$x\in V$, there exists ... 3answers 1k views What is this subgroup of$\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: ...