Chromatic Polynomials of $C_4\times P_n$ and $C_5\times P_n$

Abstract

The chromatic polynomial of a graph $G$, $P(G;\lambda )$, is the polynomial in $\lambda$ which counts the number of distinct proper vertex $\lambda$-colorings of $G$, given $\lambda$ colors. We compute $P(C_4\times P_n;\lambda )$ and $P(C_5\times P_n;\lambda )$ in matrix form and will find the generating function for each of these sequences.