5. (10 points) Given a graph $G3$, the first vertex can be colored with any color. The second one can be colored with any color that was not chosen from the first vertex. The adjacent vertices of the $G3$ are colored differently.

a. (2 points) The chromatic number is the smallest number of colors needed to produce a proper coloring of a graph. What is the chromatic number of $G$, denoted by $\chi(G)$?

b. (3 points) The chromatic polynomial is a polynomial that represents the number of distinct ways to color the vertices of a graph. What is the chromatic polynomial of $G3$, $P_G$?

c. (5 points) Let $P_G(n)$ be the number of different ways to color the vertices of $G3$ using $n$ colors, where $n \ge 0$ is an integer. What is minimum number of $P_G(n)$, where $P_G(n) > 0$?