Graphs, $\mathbb{F}_1$-schemes and virtual mixed Tate motives
In a number of recent works [6, 7] the authors have introduced and studied a functor $\mathcal{F}_k$ which associates to each loose graph $\Gamma$ -which is similar to a graph, but where edges with $0$ or $1$ vertex are allowed - a $k$-scheme, such that $\mathcal{F}_k(\Gamma)$ is largely controlled by the combinatorics of $\Gamma$. Here, $k$ is a field, and we allow $k$ to be $\mathbb{F}_1$, the field with one element. For each finite prime field $\mathbb{F}_p$, it is noted in [6] that any $\mathcal{F}_k(\Gamma)$ is polynomial-count, and the polynomial is independent of the choice of the field. In this note, we show that for each $k$, the class of $\mathcal{F}_k(\Gamma)$ in the Grothendieck ring $K_0(\texttt{Sch}_k)$ is contained in $\mathbb{Z}[\mathbb{L}]$, the integral subring generated by the virtual Lefschetz motive.