Below is the cycle graph of $\operatorname{Dih}_4$. What I don't understand is that, since $(ba)^2=a^2$, why there isn't a link between $ba$ and $a^2$, and hence of course also $a^2$ and $ba^3$? I can see that "$e$ - $ba$" is not a cycle at all, because $(ba)^2\ne e$.
a^2
/ \
a a^3
\ /
__e_____
/ / \ \
b ba ba^2 ba^3