Chromatic polynomial properties I have some properties which I do not understand how to form a proof for.
Most of them are by induction, which is not my strong point
Any help would be fantastic. 
Looking at the chromatic polynomial $$P_G(k)=P_{G-e}(k) - P_{G/e}(k)$$
Properties 
1.If G has n  vertices then $ P_g(k)$ has degree n and the coefficient of $k^n $ is 1


*The signs of $P_g(k) $ alternate

*If G has q components the smalles non-zero term of $P_g(k)$ is the term in $k^q$
 A: The proof works by strong induction on $m=|E(G)|$. We also prove that the coefficient of $k^{n-1}$ is $m$ the number of edges.
Base case : $m=0$. Then $G$ is $n$ isolated vertices. And clearly $P_G(k) = k^n$. Satisfying all results\
Induction :  Suppose the hypothesis are true for all graphs with at most $m$ edges. Let $G$ be a graph on $m+1$ edegs, $n$ vertices, $q$ components, and let $e \in E(G)$. Then


*

*$G-e$ has $m$ edges, $n$ vertices, $q$ or $q+1$ components

*$G/e$ has at most $m$ edges, with $n-1$ vertices, and $q$ components.


Using the induction hypothesis : 
\begin{align}
P_{G-e}(k) = k^n - m &k^{n-1} + \ldots - \ldots \pm bk^q \\
P_{G/e}(k) =  &k^{n-1} - \ldots + \ldots  \mp ck^q
\end{align}
With $b\geq 0$ and $c>0$. Therefore
$$ P_G(k) = k^n - (m+1) k^n + \ldots - \ldots + \ldots \pm (b+c)k^q$$
With $b+c >0$. Provind that $P_G(k)$ is monic, with alternating signs, with second coefficient $m+1$ its number of edges, and smallest coefficient the number $q$ of components.
