[Paper Review] A class of quadratic difference equations on a finite graph

Paul Baird|arXiv (Cornell University)|Sep 15, 2011

Mathematics and Applications13 references3 citations

TL;DR

This paper introduces a class of quadratic difference equations on finite graphs that generate geometric structures—such as dimension, distance, and curvature—from purely combinatorial data. It establishes a polynomial invariant and geometric spectrum, showing how holistic geometry emerges from discrete, intrinsic graph properties via orthogonal projections of regular polytopes and invariant frameworks.

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.

Motivation & Objective

• To understand how geometric properties like curvature and dimension emerge from purely combinatorial graph structures.
• To identify conditions under which non-trivial solutions exist to a class of quadratic difference equations on finite graphs.
• To define and characterize a polynomial invariant and the geometric spectrum associated with a graph.
• To explore the role of holomorphic functions and isostates as extremal solutions under a natural energy functional.
• To establish a framework where geometry arises intrinsically from graph structure without embedding in an ambient space.

Proposed method

• Formulates a quadratic difference equation at each vertex: γ(x)Δφ(x)² = (dφ)²(x), where Δφ is the Laplacian and (dφ)² is the symmetric square of the derivative.
• Introduces the geometric spectrum Σ ⊂ ℝ as the set of constant γ-values for which non-trivial solutions φ: V → ℂ exist.
• Uses orthogonal projections of regular polytopes and invariant frameworks to construct particular solutions.
• Applies Gröbner basis techniques to analyze the system of polynomial equations and derive the γ-polynomial invariant.
• Defines holomorphic mappings between graphs that preserve the structure of the equations, with dilation λ(x) quantifying local distortion.
• Linearizes the equations to study perturbations, yielding conditions like ⟨dφ, dξ⟩ₓ = 0 for holomorphic solutions.

Experimental results

Research questions

• RQ1 How can geometric invariants such as curvature and dimension be derived from purely combinatorial graph properties?
• RQ2 What conditions on a finite graph allow non-trivial solutions to the quadratic difference equation γΔφ² = (dφ)²?
• RQ3 What is the role of the geometric spectrum in encoding intrinsic geometric information?
• RQ4 How do orthogonal projections of regular polytopes and cyclic sequences give rise to solutions of the equation?
• RQ5 In what way do holomorphic mappings and isostates reflect physical principles like relationality and extremal energy?

Key findings

• The geometric spectrum Σ ⊂ ℝ is well-defined for any finite graph and encodes intrinsic geometric data, analogous to the spectrum of the Laplacian.
• Non-trivial solutions exist if and only if γ ∈ Σ, and the set Σ is finite and discrete for finite graphs.
• For holomorphic solutions (γ ≡ 0), the linearized equation reduces to ⟨dφ, dξ⟩ₓ = 0, preserving the normalization freedom φ ↦ λφ + μ.
• The γ-polynomial is a polynomial invariant of the graph, constructed from the system of equations, which encodes the geometric spectrum.
• Holomorphic mappings f: Γ → Σ preserve the equation structure, with γ(x) determined by μ(f(x)), λ(x), and m(f(x)) via γ(x) = n(x)μ(f(x))/(λ(x)m(f(x))).
• The framework realizes a relational, background-independent model of geometry, where physical quantities are relative and geometry emerges from combinatorics.