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## Analysis

A function f:A→B between two ordered sets A and B is called unimodular iff it either "goes up, then down", or "goes down, then up".

Getting technical, f is unimodular iff there exists a cut A=I∪J (a disjoint union of two intervals, such that ∀x∈I&y∈J.x<y) for which f is monotone on each of I and J.

Getting even less technical, f is unimodular iff it looks like a smiley or a frowney.

### Examples.

1. Any monotone function.
2. f(x)=x2 and -f(x).
3. cos(x) on the interval [-π,π]

## Graph Theory, Group Theory

There are many equivalent formulations of this condition. I have chosen the simplest one; however, proving its equivalence to other -- unfortunately more common -- formulations is very hard. The property is definitely esoteric, and possibly not for the faint of heart.

Let G=(V,E) be a locally finite (i.e. having finite degree at all vertices) graph which is transitive, and let Γ=Aut(G) be the group of automorphisms of G (i.e. for every x,y∈V, there is some automorphism γ∈Γ of G for which γ(x)=y). For every x∈V, denote as the stabilizer of x

Stab(x) = {γ∈Γ: γ(x)=x}
the set of all automorphisms of G fixing x. This is a subgroup of Γ. Because G is locally finite, for every y∈G the set
(Stab(x))(y)={γ(y): γ∈Stab(x)}
is finite.

For instance, if G is the two dimensional square grid, then Γ is finitely generated by the elements

• r - move one unit to the right: r(x,y)=(x+1,y);
• u - move one unit up: r(x,y)=(x,y+1);
• c - rotate clockwise around (0,0): c(x,y)=(y,-x);
• f - flip everything around x=0: f(x,y)=(-x,y).
For any v=(a,b)∈V, Stab(v) is in fact finite, and contains just the identity, the 3 rotations around v, the 2 flips about x=a and about y=b, and a rotation of each. It is D4, the dihedral group with 8 elements.

As Stab(v) is finite, clearly any u∈V can only be mapped to finitely many (in fact exactly 8, unless u=v) images by elements of Stab(v).

FINALLY, after all this preparation, we say that G is unimodular if for every x,y∈V,

|(Stab(x))(y)| = |(Stab(y))(x)|.
Each element does onto other elements as other elements do onto it.

#### Nonexamples.

Unimodularity of graphs, once grasped, appears to hold for all graphs. Here are two graphs (nodes to be added when I get a round tuit) which are not unimodular:

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