Find a number of different coloring of set
$$ V = \left\{1, ..., 6 \right\} $$ in use of $2$ colors. We treat two coloring like the same if they belong to the same orbit of the group tree automorphisms with the set of vertices V and the set of edges $$ \left\{ \left\{1,3 \right\}, \left\{2,3 \right\} , \left\{3,4 \right\}, \left\{ 4,5 \right\}, \left\{4,6 \right\} \right\} $$
Solution
Permutation $f$ is automorphism of $G$ if and only if $$ \left\{u,v \right\} \in E \text{ if and only if } \left\{f(v), f(w) \right\} \in E $$ We should find chromatic polynomial: $$W(x) = x^6 $$
so $W(2) = 64 $ Now, lets find number of automorphism (I have no experience with that before this task) $$ 1 \rightarrow 5/6 \\ 2 \rightarrow 6/5 \\ 3 \rightarrow 4/4 \\ 4 \rightarrow 3/3 \\ 5 \rightarrow 1/2 \\ 6 \rightarrow 2/1$$ so we have $2$ permutations so our answer is $$ \frac{64}{2} = 32$$ Can somebody check this solution?