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Looking at the graphs of diheral groups I noticed most have vertices of degree $2$ or $3$. The reason might seem obvious but in mathematics I have learned not to assume anything!

  1. Is there such a dihedral graph that has vertices of degree $3$?

And what about diheral symmetry in three dimensions? I have found a graph (I am sure there are many) that has vertices of degree $4$:

  • pentagonal antiprism

    1. Can we walk about the graph using generators just as we can a diheral graph?

Also, I found another graph with vertices of degree $4$:

  • regular octahedron

    1. Does this have diheral symmetry in three dimensions? And, as before, can we walk about its graph using generators just as we can a diheral graph?
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Well, instead of just saying "dihedral graph", let's break down what you're really interested in. You're asking about the degree of the undirected Cayley graph of a dihedral group with respect to some set of generators of the group. There are two variables to play with:

  1. Which dihedral group?
  2. Which set of generators?

Suppose that the group is $D_{2\times5}=\langle r, s \mid r^5, s^2, (sr)^2 \rangle$.

  • The usual way to make this into a graph is to take the Cayley graph with respect to the canonical generators $\{r,s\}$. That graph has degree $3$; every vertex is incident to edges induced by $r$, $r^{-1}$, and $s$.
  • You could also use the generators $\{s, sr\}$. That graph has degree $2$; every vertex is incident to edges induced by $s$ and $sr$. The graph is a cycle graph with $10$ vertices and alternating-color edges.
  • Or you could use the generators $\{r,s,sr\}$. That graph has degree $4$; every vertex is incident to edges induced by $r$, $r^{-1}$, $s$, and $sr$.

So the answer to your question is Yes. In fact, you should draw out the latter graph; I'm pretty sure that it is, in fact, the pentagonal antiprism! Similarly, the octahedral graph should be the Cayley graph of $D_{2\times3}=\langle r, s \mid r^3, s^2, (sr)^2 \rangle$ with respect to $\{r,s,sr\}$.

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    $\begingroup$ I think this all checks out! There's a neat webpage of especially symmetric Cayley graphs; indeed, an octahedron is listed as the Cayley graph for $D_{2 \cdot 3}$. $\endgroup$ – pjs36 Feb 23 '18 at 6:04

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