Question

Problem 7. (20 pts) Solve the following two parts. (a) For each of the six trees in Figure 3, compute their chromatic polynomial. FIGURE 3. The six graphs for Problem 7.(a). (b) Show that the chromatic polynomial $pi_T(x)$ of a tree $T = (V, E)$ is $pi_T(x) = x(x - 1)^{|V|-1}$. (c) Prove that the chromatic polynomial $pi_{K_n}(x)$ of the complete graph $K_n$ in $n$ vertices is given by the expression $pi_{K_n}(x) = x(x - 1)(x - 2)dots(x - (n - 1))$.

          Problem 7. (20 pts) Solve the following two parts.
(a) For each of the six trees in Figure 3, compute their chromatic polynomial.
FIGURE 3. The six graphs for Problem 7.(a).
(b) Show that the chromatic polynomial $pi_T(x)$ of a tree $T = (V, E)$ is
$pi_T(x) = x(x - 1)^{|V|-1}$.
(c) Prove that the chromatic polynomial $pi_{K_n}(x)$ of the complete graph $K_n$ in $n$
vertices is given by the expression
$pi_{K_n}(x) = x(x - 1)(x - 2)dots(x - (n - 1))$.