The chromatic polynomial for cycle graphs

Let $P(G,lambda)$ denote the number of proper vertex colorings of $G$ with $lambda$ colors. The chromatic polynomial $P(C_n,lambda)$ for the cycle graph $C_n$ is well-known as $$P(C_n,lambda) = (lambda-1)^n+(-1)^n(lambda-1)$$ for all positive integers $nge 1$. Also its inductive proof is widely well-known by the emph{deletion-contraction recurrence}. In this paper, we give this inductive proof again and three other proofs of this formula of the chromatic polynomial for the cycle graph $C_n$.