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.
Download