bibkey: chavez2023extremalpolynomialnorms
authors: Ángel Chávez, Sarah Fullerton, Matilda LaFortune, Keyron Linarez, Nethmin Liyanage, Justin Son and Tyler Ting
year: 2023
title: Extremal polynomial norms of graphs
doi: 10.48550/arXiv.2311.04689
url: https://arxiv.org/abs/2311.04689v3
claim: "Problem 1 asks for the complete homogeneous symmetric norms of finite path adjacency matrices in even degrees at least four."
strata_touched:
  - D5/S3/Combinatorics/Graph/PathCHS/PathCHSNormClosedForm
license: citation-only
triage: anchor

Complete homogeneous symmetric norms of paths

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DOI: 10.48550/arXiv.2311.04689

Source: https://arxiv.org/abs/2311.04689v3

Page 2, Equation (1.1): “∑{d=0}^{∞} h_d(x₁, x₂, …, x_n) t^d = ∏{i=1}^{n} 1/(1 − x_i t).”

Page 2, Theorem 1.4: “Suppose d ≥ 2 is even. The following function defines a norm on H_n: ‖A‖_d = h_d(λ₁, λ₂, …, λ_n)^{1/d}.”

Page 14, Section 5, Problem 1: “The path of order n has m = n − 1 edges. Proposition 5.14 (a) therefore implies that ‖P_n‖₂ = √(n − 1). However, computing ‖P_n‖_d for d ≥ 4 seems unattainable at the moment. We leave this is an interesting open problem. Problem 1. Compute ‖P_n‖_d for d ≥ 4.”

Vertices are indexed by Fin n. The adjacency matrix has a one precisely at indices whose natural coordinates differ by one. Complete homogeneous symmetric polynomials are evaluated on its real eigenvalues; the norm is the real power with exponent 1/d. For d=2m the expression uses zero-extended binomial coefficients and a natural quotient m/(n+1).

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