References request: which semigroups give Cayley graphs which are different from the Cayley graphs given by groups? I would like to know which semigroups give Cayley graphs which are different from the Cayley graphs given by groups. Are there some references? 
For example, do the Cayley graphs of complete simple semigroups give new examples of Cayley graphs which are not given by the Cayley graphs of groups?
Thank you very much.
 A: I am not an expert, but here is what I would do. Take the (additive) semigroup $S$ of nonnegative integers with the generator $1\in S$. Then, clearly, the Cayley graph of $S$ is the half-line. On the other hand, clearly, half-line cannot be a Cayley graph of a group. 
Edit: Since you are apparently interested in "completely simple" semigroups, note that this example is also completely simple. Furthermore, simple google search 
"Cayley graph of semigroup" yields many references. It would help, therefore, if you did some searching yourself and then stated a more focused question.  
A: Maybe this paper will be useful for you:
A. V. Kelarev and S. J. Quinn
A Combinatorial Property and Cayley Graphs
of Semigroups, Semigroup Forum, v.66, 2003, 89-96.
A: You might be interested in the thesis of Simon Craik, which can be found here. He studied the "ends" of a semigroup, and this analysis yields many differences between the Cayley graphs of groups and of semigroups. The definition of the "number of ends", which is what we care about, can be interpreted as "Draw the Cayley graph and take out increasingly bigger and bigger chunks from the middle. How many connected components will you be left with?" For example, finite groups have zero ends, the group $\mathbb{Z}$ has two ends, the group $\mathbb{Z}\times\mathbb{Z}$ has one end and finally the free group on two generators $F_2$ has infinitely many ends. Indeed, a finitely generated group can have either zero, one, two or infinitely many ends, and Stallings' classified these cases. However, the Cayley graph of a semigroup can have an arbitrary number of ends, such as 6 or 12412. See Section 3.1 of Craik's thesis for the relevant example. More formally, he proves that for every pair of integers $m$ and $n$ there exists a semigroup $S_{(m, n)}$ such that the left Cayley graph has $m$ ends while the right Cayley graph has $n$ ends.
(In fact, it isn't clear that the semigroups $S_{(m, n)}$ are not completely simple. I wonder if they are. This would clearly answer your question. You might want to investigate...)
