Find the Chromatic Polynomial of the graph given below

Question

I want to find the chromatic polynomial for this graph:

My polynomial was x(x-1)((x-2)^2) but i am not sure that this is correct.

Answer 1

Your approach and answer is correct, the graph is a 2-clique sum of two $K_3$'s and thus the chromatic polynomial $\frac{P(K_3,x)P(K_3,x)}{x(x-1)}$.

I offer an alternative solution. We apply the deletion-contraction formula in this scenario.

We let your graph be $G$. Removing the diagonal $xv$, we obtain $P(G,x)=P(G-xv,x)- P(G/xv,x)=P(C_4,x)-P(P_3,x)$.

The characteristic polynomial for cycles and paths are well known.

Thus, $P(G,x)= (x-1)^4+(-1)^4(x-1)-x(x-1)^2=x(x-1)(x-2)^2$.