Figure 1

The complete graph$\mathit{G}\mathbf{=}{\mathit{K}}_{\mathbf{2}}$has$\mathcal{A}\mathbf{=}{\mathit{S}}_{\mathbf{2}}$as automorphism group. All transformations T have Lefschetz number 1. The zeta function of $T=\mathrm{Id}$ only involves $a(1)=1$, $b(1)=2$ so that ${(1-z)}^{1-2}$. The reflection T has a fixed $K_2$ of negative signature giving $c(1)=1$ and a 0-dimensional periodic point of period 2 giving $b(2)=1$ so that $\zeta (z)=(1+z)/(1-z^2)=1/(1-z)$.