Is there a closed form for the characteristic polynomial of the graph cycle (of $n$ edges and $n$ summits)? 


I know it for the graph path (it is a Chebyschev polynomial), but I couldn't find a closed form when adding the missing edge.



Is there a closed form for  the characteristic polynomial of the graph cycle (of n edges and n summits) ?
If I use $\phi$ for characteristic polynomial, then
$$
\phi(C\_n,t) = \phi(P\_n,t) - \phi(P\_{n-2},t) - 2.
$$
This follows from the formulas in Section 4.1 of "Algebraic Combinatorics" by yours truly, 
see in particular Exercise 5 in Chapter 4. (Sorry about the self-referencing, but I have
no recollection of seeing the formula elsewhere.) Since the coefficient of $t^{n-2r}$ in $\phi(P\_n,t)$ is $(-1)^r\binom{n-r}{r}$, the above identity leads to a simple expression for the coefficients of $\phi(C\_n,t)$.


