Finite graphs and configurations of points

TL;DR

This paper extends the Atiyah problem on point configurations and related conjectures to a broader setting involving finite graphs, introducing a new geometric inequality and a generalized amplitude function.

Contribution

It introduces a novel generalization of the Atiyah determinant called the G-amplitude function, applicable to finite graphs, and connects it to existing conjectures in a new framework.

Findings

Definition of the G-amplitude function for finite graphs

Recovery of Atiyah--Sutcliffe conjectures when G is a complete graph

Proposal of new geometric inequalities involving point configurations

Abstract

We generalize the Atiyah problem on configurations and the related Atiyah--Sutcliffe conjectures 1 and 2 using finite graphs, configurations of points and tensors. Our conjectures are intriguing geometric inequalities, defined using the pairwise directions of the configuration of points, just as in the original problem. The generalization of the Atiyah determinant to our setting is no longer a determinant. We call it the $G$-amplitude function, where $G$ is a finite simple graph, in analogy with probability amplitudes in quantum physics. If $G = K_n$ is the complete graph with $n$ vertices, we recover the Atiyah--Sutcliffe conjectures 1 and 2.