In answering a recent Question about $6$-regular graphs with $42$ vertices, I commented that even the vertex-transitive cases afford a lot of choices. Three such graphs were proposed, and I left it as a challenge to the interested Reader to verify non-isomorphism.
My hint was to fix a vertex $v$ and count the number of edges incident only to the six neighbors of $v$. The three graphs gave respective counts of $9,15,6$ edges between such neighbors, proving them non-isomorphic. A fourth example can be adduced, that of a $6$-regular vertex-transitive bipartite graph, where obviously the number of edges between neighbors of any one vertex is zero.
My question is what are the possible counts of edges between neighbors in $6$-regular vertex-transitive graphs having $42$ nodes. All the examples described have counts that are multiples of three, which I speculate is a necessary but not sufficient condition.
I have some thoughts about attacking this problem which I'd like to share.
A nontrivial graph $G$ which is vertex transitive will have a nontrivial automorphism group $\mathscr G$. Often the study of $G$ can be illuminated by the study of $\mathscr G$, as we see in this recent arXiv.org paper (2016) by Jing Chen AND Binzhou Xia:
Note $\mathscr G$ has at least a subgroup $\mathscr H$ generated by those automorphisms which map some fixed vertex $v$ to one of its neighbors. If $G$ is connected, then $G$ is $\mathscr H$-vertex-transitive since we can get from $v$ to any other vertex $u$ by a sequence of automorphisms in $\mathscr H$. There is a slightly subtle concept involved here, noting that an automorphism $h$ maps $v$ to one of its neighbors if and only if $h$ maps every $u\in G$ to one of its neighbors, so that every step in the sequence outlined can be taken by one of the generators of $\mathscr H$.
If $G$ is not connected, then the components of $G$ must be equal in size, and in order for $G$ to be $6$-regular, that size must be a proper divisor $d$ of $42$ which is at least $7$. In other words the sizes of nontrivial components are tightly restricted, and we hope these possibilities can be quickly exhausted.
Update: In checking the known vertex transitive connected graphs of regular degree $6$, I found examples with $21$ nodes where the neighbors of a fixed vertex share a number of edges not divisible by $3$ (namely $4$ or $7$ edges are shared).