A Spectral Lower Bound on Chromatic Numbers using $p$-Energy

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.