A Spectral Lower Bound on Chromatic Numbers using $p$-Energy
Clive Elphick, Quanyu Tang, Shengtong Zhang
TL;DR
This work introduces a unified framework of $p$-energies, decomposing a graph's adjacency spectrum into positive and negative parts to derive a family of lower bounds for $\chi(G)$ and its relaxations. By combining stochastic block decompositions, Haar-measure averaging, and majorization techniques, the authors prove that for all $p\ge 0$, $\xi_f(G) \ge 1 + \max\left\{ \mathcal{E}_p^+(G)/\mathcal{E}_p^-(G), \mathcal{E}_p^-(G)/\mathcal{E}_p^+(G) \right\}^{1/|p-1|}$, which tightens and unifies the classical bounds (Hoffman, Ando–Lin, Elphick–Wocjan) at $p=\infty,2,0$ respectively. The approach uses a novel synthesis of linear algebra and measure theory, including stochastic block decompositions, majorization, and the Haar measure, to extend the analysis to non-integer $p$, yielding sharper bounds in several graphs such as the Tilley graph and a class of graphs where the optimal bound occurs at non-integer $p$. The results also resolve two conjectures by Elphick and Wocjan and imply tightness for balanced Turán graphs, highlighting the practical impact for estimating chromatic parameters in spectral graph theory.
Abstract
Let $A_G $ be the adjacency matrix of a simple graph $ G $, and let $ χ(G) $, $ χ_f(G) $, $ χ_q(G) $, $ ξ(G) $ and $ ξ_f(G) $ denote its chromatic number, fractional chromatic number, quantum chromatic number, orthogonal rank and projective rank, respectively. For $ p \geq 0 $, we define the positive and negative $ p $-energies of $ G $ by $$ \mathcal{E}_p^+(G) = \sum_{λ_i > 0} λ_i^p, \quad \mathcal{E}_p^-(G) = \sum_{λ_i < 0} |λ_i|^p, $$ where $ λ_1 \geq \cdots \geq λ_n $ are the eigenvalues of $A_G $. We prove that for all $ p \geq 0 $, $$ χ(G) \geq \left\{χ_f(G), χ_q(G), ξ(G) \right\} \geq ξ_f(G) \geq 1 + \max\left\{ \frac{\mathcal{E}_p^+(G)}{\mathcal{E}_p^-(G)}, \frac{\mathcal{E}_p^-(G)}{\mathcal{E}_p^+(G)} \right\}^{\frac{1}{|p - 1|}}. $$ This result unifies and strengthens a series of existing bounds corresponding to the cases $ p \in \{0, 2, \infty\} $. In particular, the case $ p = 0 $ yields the inertia bound $$ χ_f(G) \geq ξ_f(G) \geq1 + \max\left\{\frac{n^+}{n^-}, \frac{n^-}{n^+}\right\}, $$ where $ n^+ $ and $ n^- $ denote the number of positive and negative eigenvalues of $ A_G $, respectively. This resolves two conjectures of Elphick and Wocjan. We also demonstrate that for certain graphs, non-integer values of $ p $ provide sharper lower bounds than existing spectral bounds. As an example, we determine $ χ_q $ for the Tilley graph, which cannot be achieved using existing (unweighted) $p$-energy bounds. Our proof employs a novel synthesis of linear algebra and measure-theoretic tools, which allows us to surpass existing spectral bounds.