Chromatic polynomial of a unicyclic graph ($p_G(x)=(x-1{)}^n+(-1{)}^{\ell }(x-1{)}^{n-\ell +1}$)

If a unicyclic graph has $n$ vertices and its unique graph cycle has length $\ell$, each of its $n-\ell$ off-graph cycle vertices contributes a factor $x-1$ after its parent in a rooted tree is coloured. Thus $p_G(x)=(x-1{)}^{n-\ell }{p}_{C_{\ell }}(x)$. Substituting the chromatic polynomial of a cycle yields the displayed formula, equivalently by repeated application of the chromatic polynomial after attaching a leaf.