[Paper Review] Sums of squares of eigenvalues and the vector chromatic number

Gabriel Coutinho, Thomás Jung Spier|arXiv (Cornell University)|Aug 8, 2023

Graph theory and applicationsMathematics3 citations

TL;DR

This paper proves a conjecture by Wocjan, Elphick, and Anekstein, establishing that the minimum of the sums of squares of positive or negative eigenvalues of a graph's adjacency matrix is lower bounded by twice the number of edges divided by the vector chromatic number. The proof leverages a novel semidefinite programming formulation of the vector chromatic number and applies matrix decomposition and Cauchy-Schwarz inequalities to derive the bound.

ABSTRACT

In this short paper we prove that the sum of the squares of negative (or positive) eigenvalues of the adjacency matrix of a graph is lower bounded by the sum of the degrees divided by the vector chromatic number, resolving a conjecture by Wocjan, Elphick and Anekstein (2018).

Motivation & Objective

• To resolve a conjecture by Wocjan, Elphick, and Anekstein (2018) on the lower bound of sums of squares of eigenvalues in terms of the vector chromatic number.
• To strengthen prior results by Ando and Lin (2018) and Guo and Spiro (2020) on spectral bounds in graph theory.
• To provide a new, direct proof that avoids reliance on homomorphisms to edge-transitive graphs by exploiting structural properties of the vector chromatic number's SDP formulation.
• To establish a tight spectral inequality connecting the sum of squared positive or negative eigenvalues to the vector chromatic number and the number of edges.

Proposed method

• Formulates the vector chromatic number using a novel semidefinite program (SDP) with constraint ⟨I + A̅, Z⟩ = 1, where A̅ is the complement's adjacency matrix.
• Introduces a key inequality: ⟨J, Z⟩ ≤ χ_vec(G)⟨J - A, Z⟩ for all Z ≥ 0 and Z ⪰ 0, derived from the SDP formulation.
• Applies Lemma 3, a matrix decomposition inequality, to relate the squared norms of A⁺ and A⁻ under orthogonality and edge-complement constraints.
• Uses the spectral decomposition A = A⁺ - A⁻, where A⁺ and A⁻ are positive semidefinite projections onto positive and negative eigenspaces.
• Applies Cauchy-Schwarz to the off-diagonal entries of A⁺ and A⁻ on non-edges, linking the sum of squares to the vector chromatic number.
• Establishes the bound min{s⁺, s⁻} ≥ 2m / χ_vec(G) by symmetry, using the roles of A⁺ and A⁻ interchangeably.

Experimental results

Research questions

• RQ1 Is the sum of the squares of the positive or negative eigenvalues of a graph’s adjacency matrix bounded below by 2m / χ_vec(G), as conjectured by Wocjan, Elphick, and Anekstein?
• RQ2 Can the vector chromatic number be re-expressed via a new SDP formulation that simplifies spectral bounds involving eigenvalue sums?
• RQ3 Does the absence of homomorphism assumptions in the proof lead to a more general or robust inequality for spectral graph invariants?
• RQ4 Can the interplay between positive and negative eigenspaces be formalized through matrix decomposition and SDP duality to yield tight spectral bounds?
• RQ5 Is the inequality min{s⁺, s⁻} ≥ 2m / χ_vec(G) tight across all graph classes, and what are its extremal cases?

Key findings

• The sum of the squares of the positive eigenvalues, s⁺, and the sum of the squares of the negative eigenvalues, s⁻, satisfy min{s⁺, s⁻} ≥ 2m / χ_vec(G), where m is the number of edges.
• The bound is proven using a new SDP formulation of the vector chromatic number, which allows a direct derivation without relying on graph homomorphisms.
• The proof establishes that s⁺ ≤ (χ_vec(G) - 1)s⁻ and s⁻ ≤ (χ_vec(G) - 1)s⁺, which together imply the main inequality.
• The result strengthens earlier bounds by Ando and Lin (2018) and Guo and Spiro (2020), providing a tighter spectral constraint.
• The key technical insight is the use of matrix decomposition and Cauchy-Schwarz on off-diagonal entries to relate the squared norms of A⁺ and A⁻.
• The bound is symmetric and tight in the sense that equality holds when the spectral and chromatic structures align, such as in certain strongly regular graphs.