An automorphism of a graph $\displaystyle G$ is an isomorphism between $\displaystyle G$ and $\displaystyle G$ itself. How many

automorphisms does the following (labelled) graph have:

$\displaystyle C_n + e$, i.e. a cycle with $\displaystyle n$ vertices and a unique chord (i.e. with a unique edge

connecting two non-consecutive vertices of the cycle)?