# Cayley graph on $D_{2n}$ and $\mathbb Z_n$

How we can make Cayley graph on $D_{2n}$ and $\mathbb Z_n$? What can be S in $Cay(D_{2n},S)$ and $Cay(\mathbb Z_n ,S)$, Please write one example. Thanks for advise.

• Well...what have you tried? – Elchanan Solomon Sep 21 '13 at 13:00

Here's an example of a Cayley graph for the group $\mathbb{Z}_{10}$:

The vertices are the underlying set of the group $\mathbb{Z}_{10}$, and we draw edges between $g$ and $g+h$ whenever $g,h \in S$ where $S=\{\pm 1, \pm 2\}$.

Here's an example of a Cayley graph for the dihedral group $D_{10}$:

The vertices are the underlying set of the dihedral group with presentation $$D=\langle f,r | f^2=e, r^5=e, rf=fr^{-1} \rangle,$$ (we can think of $f$ as "flip" and $r$ as "rotation") and we draw edges between $g$ and $g+h$ whenever $g,h \in S$ where $S=\{f,r,r^{-1}\}$.

• thanks, but can you give me an example for 3-hypergraph cayley on $D_{2n}$? – mehranian Sep 21 '13 at 13:31
• A 3-uniform Cayley hypergraph for $D_{2n}$, that'd be more difficult to draw. I'm not even sure of the definition in this case, but I expect if we take one hyperedge $\{g_1,g_2,g_3\}$, then generate the remaining edges as $\{hg_1,hg_2,hg_3\}$ for all $h \in D_{2n}$, this will give an example. – Rebecca J. Stones Sep 21 '13 at 13:36

Considering the definition of Cayley graph done by @Owen, you can do it as follows:

For $\mathbb Z_n=\langle a\rangle=\{1,a,a^2,\cdots a^{n-1}\}$:

And for $D_{2n}$, I did it for $D_6=\{1,a,a^2,b,ab,a^2b\}$ of order $6$ instead:

• Nothing like paper and pencil (or pen)! +1 – Namaste Sep 21 '13 at 13:49
• @amWhy: Pen,Amy. :-) – mrs Sep 21 '13 at 13:58

Recalling the definition of Cayley graph of a group $G$, remember that the vertex set is the set $G$ and two vertices, $g, h\in G$, are connected by an edge if there exists some $a\in S$ such that $g^{-1}h\in S$. So this means that $S\subset G$. Now in principal you could let $S$ be any subset of $G$ but usually (always?) you want to choose $S$ to be a generating set for $G$ because this ensures that the resulting graph is connected. So in paticular, since groups have many different generating sets, the resulting graphs will be different. So there is no such thing as $\textbf{the}$ Cayley graph of a group.

Just to give one example. If $G=\{e, g_1, ..., g_n\}$, note $|G|=n+1$, and if we let $S=G\setminus\{e\}$. Then Cayley($G, S)$= $K_{n+1}$, the complete graph on $n+1$ vertices.

However, in general you do not do this. Instead, you try to use the smallest $S$ possible. In particular for the case $\mathbb{Z}_n$ you can choose $S=\{1\}$. In this case you will get that the Cayley graph is a loop with $n$ points on it.

• thanks, but you can give me an example of 3-hypergraph cayley on $D_{2n}$? – mehranian Sep 21 '13 at 14:05