Is there a dihedral graph in which the vertices have degree 4? 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! 


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*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 $4$:


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*pentagonal antiprism


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



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


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*regular octahedron 


*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?


 A: 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:


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*Which dihedral group?

*Which set of generators?


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


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*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 $\{r,s,sr\}$.
