Automorphisms of Deitmar schemes, I. Functoriality and Trees

In a recent paper [3], the authors introduced a map $mathcal{F}$ which associates a Deitmar scheme (which is defined over the field with one element, denoted by $mathbb{F}_1$) with any given graph $Gamma$. By base extension, a scheme $mathcal{X}_k = mathcal{F}(Gamma) otimes_{mathbb{F}_1} k$ over any field $k$ arises. In the present paper, we will show that all these mappings are functors, and we will use this fact to study automorphism groups of the schemes $mathcal{X}_k$. Several automorphism groups are considered: combinatorial, topological, and scheme-theoretic groups, and also groups induced by automorphisms of the ambient projective space. When $Gamma$ is a finite tree, we will give a precise description of the combinatorial and projective groups, amongst other results.