A class of quadratic difference equations on a finite graph

2011/09/15 by Paul Baird, Baird, Paul
Mathematics · Physics and Astronomy · #05C10 #39A14 #52B11 #52C99 #Advanced Differential Equations and Dynamical Systems #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematics and Applications #Point processes and geometric inequalities #math-ph #math.CO #math.MP #msc:05C10 #msc:39A14 #msc:52B11 #msc:52C99

paper · pdf · doi:10.48550/arxiv.1109.3286

arxiv created 2011/09/15 · openalex publication_date 2011/09/15 · arxiv updated 2011/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study a class of complex polynomial equations on a finite graph with a view to understanding how holistic phenomena emerge from combinatorial structure. Particular solutions arise from orthogonal projections of regular polytopes, invariant frameworks and cyclic sequences. A set of discrete parameters for which there exist non-trivial solutions leads to the construction of a polynomial invariant and the notion of a geometric spectrum. Geometry then emerges, notably dimension, distance and curvature, from purely combinatorial properties of the graph.

Citations

• A generalized inverse for matrices
• Relational Quantum Mechanics
• The volume formulas for a spherical tetrahedron

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