More Graph Geometry

I tried pushing on the thought I had before and seeing what ideas from algebraic geometry I could transport by replacing commutative rings with graphs. Still not sure it's particularly useful, but it's a fun exercise!

As a reminder, the setting is undirected graphs with no loops, and no multiple edges. The main intuition I'm squeezing is trying to pay attention to when we care, in algebraic geometry about the two concepts of equality with zero and invertibility and to replace them with merging two vertices and putting an edge between two vertices in a graph. We don't have any distinguished notion of zero among the vertices of a graph, so equality with zero gets replaced by equality between any two vertices, and invertibility (which is a kind of "evidence of apartness from zero") is replaced by some kind of evidence for apartness, which I'm choosing to represent as graph edges

The fact that edges are supposed to mean apartness/disjointness is reflected by the fact that I demanded there aren't any loop edges in the graphs.

The Dictionary So Far

Here's an attempt at building up a little dictionary of familiar concepts in AG and some alleged analogues in graph theory. The first question I started with was, ok, what do prime ideals of a ring become? It makes sense to me to see a prime ideal as saying "I want to force these elements (the elements in the ideal) to be equal to zero, and all other elements to be invertible".

I figured maybe the graph analog of that is "consistent partition", i.e. a partition of its vertex set into blocks such that no edge connects two vertices in the same block. It's a set of equations we can consistently force, without running afoul of any edge that asserts inequality, and also assert inequality of everything else

Algebraic Geometry	Graph Theory
Commutative ring $R$	Graph $G$
Ring hom $R \to S$	Graph hom $G \to H$
Ideal $\mathfrak{a} \subseteq R$	Set of potential edges $S$ on vertex set $V(G)$
Quotient $R/\mathfrak{a}$	Contraction $G/S$
Localization $R[f^{-1}]$	Edge addition $G \cup \{e\}$
Prime ideal $\mathfrak{p} \subseteq R$	Consistent partition $K$ of $G$
$\mathfrak{p}_1 \supseteq \mathfrak{p}_2$ (specialization)	$K_1$ is coarser than $K_2$
Generic point $(0)$	Discrete partition (every vertex in its own block)

These partitions can be seen as colorings, except quotiented out by permutations of the coloring set. This means that the natural analog of a maximal ideal is a graph minimal in this specialization order, and is a way of identifying a minimal coloring of a graph. Since we can build up a chain of progressively coarser and coarser partitions on the way to a maximal ideal, the "Krull dimension" of a graph has the same information as its chromatic number! That's kind of neat.

Algebraic Geometry	Graph Theory
Maximal ideal	Minimal coloring (can't merge any two blocks)
Krull dimension	$\|V(G)\| - \chi(G)$

I think the thing that plays the role of fields is complete graphs. In a field everything that can be invertible (which excludes zero) is invertible, and in a complete graph, every edge that can exist, (which excludes self-edges) does exist. The "residue field" of a "prime ideal" is what we get by contracting every required contraction in the partition, and adding edges everywhere else: it's the complete graph on the set of blocks in the partition. The local ring $R_\mathfrak{p}$ is what we get if we invert everything outside the prime ideal, so in graphs it should be what we get if we add edges between any two vertices that aren't in the same block, so a complete multipartite graph. Working "over a field $\mathbb{k}$" becomes working over

Algebraic Geometry	Graph Theory
Field	Complete graph $K_n$
Residue field $\kappa(\mathfrak{p})$	$K_{\|K\|}$ (the complete graph on as many vertices as $K$ has blocks)
Local ring $R_\mathfrak{p}$	Complete multipartite graph arising from $K$

The Spectrum

We can try to recapitulate the construction of the spectrum as a locally ringed space in AG, instead constructing a "locally graphed space".

Given a graph $G$, define $\mathsf{Spec}(G)$ as a topological space whose points are consistent partitions of $G$. The topology is generated by basic closed sets: for each pair of vertices $i, j$, $$V(i\text{---} j) = \{ K \in \mathsf{Spec}(G) : K \text{ places } i \text{ and } j \text{ in the same block} \}$$ The complementary basic open sets are $$D(i\text{---}j) = \{ K \in \mathsf{Spec}(G) : K \text{ places } i \text{ and } j \text{ in different blocks} \}$$ The open set $D(i\text{---}j)$ is the locus where the inequality demanded by the edge $i\text{---}j$ is respected, analogous to the Zariski basic open $D(f)$, the locus where $f$ is nonzero.

The Structure Sheaf

For any open $U \subseteq \mathsf{Spec}(G)$, define $\mathcal{O}(U)$ to be the graph on vertex set $V(G)$ with edge set $$E(G) \;\cup\; \{ i\text{---}j : \text{every partition in } U \text{ separates } i \text{ and } j \}$$ Smaller opens have more edges, just as smaller opens in Zariski have more invertible elements. Restriction maps are edge-inclusion homomorphisms. This is a sheaf: an edge belongs to $\mathcal{O}(U)$ iff it belongs to $\mathcal{O}(U_\alpha)$ for every element of a cover.

Global sections recover the graph, as we'd expect: $\mathcal{O}(\mathsf{Spec}(G)) = G$.

Stalks are complete multipartite graphs. The stalk $\mathcal{O}_K$ at a coloring $K$ is the complete multipartite graph on the blocks of $K$.

The Chromatic Polynomial as Point-Counting

A graph homomorphism $G \to K_k$ is exactly a $k$-coloring of $G$, a function to $k$ colors. So if we instead adopted a functorial view of $\mathsf{Spec}$, we'd have a nice expression for the chromatic polynomial of $G$: $$P(G, k) = |\mathsf{Hom}(G, K_k)| = |\mathsf{Spec}(G)(K_k)|$$ The chromatic polynomial is the point-counting function evaluated at the "field" $K_k$, which seems analogous to $|\mathsf{X}(\mathbb{F}_q)|$ for a scheme $X$.