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)? 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. 


If I use $\phi$ for characteristic polynomial, then $$ \phi(C_n,t) = \phi(P_n,t)  \phi(P_{n2},t)  2. $$ 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 selfreferencing, but I have no recollection of seeing the formula elsewhere.) Since the coefficient of $t^{n2r}$ in $\phi(P_n,t)$ is $(1)^r\binom{nr}{r}$, the above identity leads to a simple expression for the coefficients of $\phi(C_n,t)$. 

