Maxima for Graphs and a New Proof of a Theorem of Turán

TLDR

The paper investigates the maximum of a square‑free quadratic form over the simplex S in ℝⁿ, a question inspired by a problem posed by J. E. MacDonald Jr.

Abstract

Maximum of a square-free quadratic form on a simplex. The following question was suggested by a problem of J. E. MacDonald Jr. (1): Given a graph G with vertices 1, 2, . . . , n. Let S be the simplex in E n given by x i ≥ 0, Σ x i = 1. What is