${\displaystyle {}G}$

${\displaystyle {}k\geq 0}$

${\displaystyle {}P_{G}\left(k\right)\geq 1\,.}$

${\displaystyle {}n={\#\left(V\right)}}$

${\displaystyle {}k(k-1)\cdots (k-n+1)\leq P_{G}\left(k\right)\leq k^{n}\,.}$

${\displaystyle {}k\geq {\#\left(V\right)}+1=n+1}$

${\displaystyle {}P_{G}\left(k\right)=\sum _{j=0}^{n}{\left(\sum _{\ell =j}^{n}(-1)^{\ell -j}{\binom {k}{\ell }}{\binom {\ell }{j}}\right)}P_{G}\left(j\right)\,.}$

Insbesondere ist das chromatische Polynom durch die Werte ${\displaystyle {}P_{G}\left(0\right),P_{G}\left(1\right),\ldots ,P_{G}\left(n\right)}$ festgelegt.