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In a number of recent works [6, 7] the authors have introduced and studied afunctor $\mathcal{F}_k$ which associates to each loose graph $\Gamma$ -which issimilar 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 thecombinatorics 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 $\mathcal{F}_k(\Gamma)$ ispolynomial-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 virtualLefschetz motive.

Graphs, $\mathbb{F}_1$-schemes and virtual mixed Tate motives