Let $G$ be a group and $S \subseteq G$ be a generating set of $G$.
The Cayley digraph of $G$ with respect
to $S$, $X=\overrightarrow {\operatorname{Cay}}(G, S)$ is a graph whose vertices are the elements of $G$ and there is an edge from $g$ to $gs$
whenever $g \in G$ and $s \in S$.
The Cayley graph, $X = \operatorname{Cay}(G, S)$ is the undirected graph whose vertices
are the elements of $G$ and there is an edge from $g$ to $gs$ and from $g$ to $gs^{-1}$ whenever $g \in G$ and $s \in S$.
In a group, an element of order $2$ is known as an "involution". i.e. a non-identity element
whose square is the identity element.
So by thinking about the above definitions I have written the following about the degree of a Cayley graph, $X=\operatorname{Cay}(G,S)$ of a group $G$, with respect to a generator set $S$:
"$\deg(X)= 2\vert S \vert $, if $S$ has elements which are not involutions"
I would like to know whether above sentence is correct. Is it ok, to mention as "$S$ has elements which are not involutions"?
Best Answer
The answer may depend on definitions of a Cayley graph and vertex degree. Namely, if we consider the directed Cayley graph then in-degree of its each vertex equals to its out-degree and equals to $|S|$. If we consider the undirected Cayley graph then we assume that $S$ is symmetric and without the identity $\{e\}$ and then degree of its each vertex equals $|S|$.