I have the following Cayley graph: $ ...\cdot \leftrightarrow \cdot \leftrightsquigarrow \cdot \leftrightarrow \cdot \leftrightsquigarrow \cdot \leftrightarrow ...$
Here are my thoughts: we have two generators $a,b$. And $a^2=b^2=e$ since we have arrows going out and back from $e$ to $a$ and $b$. So this group is the group of words $ababa...aba$ which can start and end with either $a$ or $b$. Is it the same as a free group with two relations: $\langle a, b\ |\ a^2, b^2\rangle$? Is my interpretation correct?