On Cayley graphs: Part 1
A Cayley graph can be considered as an attempt to geometrize or visualize groups . Algebraically, a group is just a bunch of letters some of them having an inverse script. By visualizing groups, we may observe some patterns in their structure. So let's see how can we construct Cayley graphs. Take an arbitrary group $G$ with a set of generators $S\subset G$. In other words, every element of $G$ can be written as: \begin{equation} g = s_1^{\epsilon_1}s_2^{\epsilon_2}\ldots s_n^{\epsilon_n},\quad \epsilon_i\in\{\pm1 \} \end{equation} As for $S$, we assume that $S^{-1} = S$, i.e. $s\in S$ implies $s^{-1}\in S$, and $1\not\in S$. In fact these assumptions are not necessary, but tend to make life a bit easier. Let us construct a Cayley graph $\Gamma=(V,E)$ as follows: $V=G$ $(g,h)\in E$ if $gs = h$ for some $s\in S$, or equivalently $g^{-1}h\in S$ We can observe few things straight away about $\Gamma$. It is undirected as $gs = h$ happens when $g = hs^{-1}$. Any two vert...