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Finite graphs and configurations of points

Joseph Malkoun

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.

Finite graphs and configurations of points

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 -amplitude function, where is a finite simple graph, in analogy with probability amplitudes in quantum physics. If is the complete graph with vertices, we recover the Atiyah--Sutcliffe conjectures 1 and 2.

Paper Structure

This paper contains 9 sections, 14 theorems, 94 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The amplitude function associated to a finite simple graph
  3. Conjectures
  4. A conjecture in matrix analysis
  5. Proof of Theorem \ref{['main-thm']}
  6. Numerical Simulations
  7. Amplitudes as tensor network contractions
  8. Conclusion
  9. Declarations

Key Result

Proposition 2.1

For any permutation $\sigma$ of $V$, we have It then follows that

Figures (5)

  • Figure 1: A finite simple graph $G$ with $7$ vertices and $6$ edges.
  • Figure 2: The complete graph $K_5$ with $5$ vertices.
  • Figure 3: A finite tree with $8$ vertices and $4$ levels
  • Figure 4: A star graph
  • Figure 5: A "linear" graph

Theorems & Definitions (23)

  • Proposition 2.1
  • Proposition 2.2
  • Corollary 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • Corollary 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Conjecture A
  • ...and 13 more