Finite graphs and configurations of points

2025/08/19 by Joseph Malkoun, Malkoun, Joseph
Computer Science · Engineering · #15A69 (Secondary) #51M15 (Primary) 05Cxx #Advanced Theoretical and Applied Studies in Material Sciences and Geometry #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #Digital Image Processing Techniques #FOS: Mathematics #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.2508.13472

openalex publication_date 2025/08/19 · openalex created_date 2025/10/09 · openalex updated_date 2026/07/28

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.

Citations

• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States

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