A connection between the $A_α$-spectrum and the Lovász theta number

TL;DR

The paper studies the smallest $\alpha$ for which the matrix $A_\alpha = \alpha D + (1-\alpha) A$ becomes positive semidefinite, revealing a tight connection to the Lovász theta number via $\alpha_0 \ge 1/\vartheta(\overline{G})$. By allowing weighted variations in $A$ and $D$, the authors show $\alpha_0 = 1/\vartheta(\overline{G})$ in this generalized setting and connect this framework to Motzkin–Straus’ quadratic formulation for $\omega(G)$. They further relate $\alpha_0$ to SDP relaxations for max-$k$-cut, yielding bounds that recover $1/\chi(G)$ and extend classical results of Nikiforov–Rojo. A copositive relaxation is introduced, giving $1/\omega(G) \le \alpha_0^{\mathcal{C}} \le 1/2$, with a constructive link to Motzkin–Straus but leaving the exact equality with $1/\omega$ as an open problem. Overall, the work bridges spectral, semidefinite, and copositive formulations to tie the $A_\alpha$-spectrum to fundamental graph parameters.

Abstract

We show that the smallest $α$ so that $αD + (1-α)A \succcurlyeq 0$ is at least $1/\vartheta(\overline{G})$, significantly improving upon a result due to Nikiforov and Rojo (2017). In fact, we display an even stronger connection: if the nonzero entries of $A$ are allowed to vary and those of $D$ vary accordingly, then we show that this smallest $α$ is in fact equal to $1/\vartheta(\overline{G})$. We also show other results obtained as an application of this optimization framework, including a connection to the well-known quadratic formulation for $ω(G)$ due to Motzkin and Straus (1964).

Theorems & Definitions (16)

• Proposition 1
• proof
• Proposition 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Proposition 5
• proof