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

Let $A_G $ be the adjacency matrix of a simple graph $ G $, and let $ \chi(G) $, $ \chi_f(G) $, $ \chi_q(G) $, $ \xi(G) $ and $ \xi_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 $$ \mathcal{E}_p^+(G) = \sum_{\lambda_i > 0} \lambda_i^p, \quad \mathcal{E}_p^-(G) = \sum_{\lambda_i < 0} |\lambda_i|^p, $$ where $ \lambda_1 \geq \cdots \geq \lambda_n $ are the eigenvalues of $A_G $. We prove that for all $ p \geq 0 $, $$ \chi(G) \geq \left\{\chi_f(G), \chi_q(G), \xi(G) \right\} \geq \xi_f(G) \geq 1 + \max\left\{ \frac{\mathcal{E}_p^+(G)}{\mathcal{E}_p^-(G)}, \frac{\mathcal{E}_p^-(G)}{\mathcal{E}_p^+(G)} \right\}^{\frac{1}{|p - 1|}}. $$ 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 $$ \chi_f(G) \geq \xi_f(G) \geq1 + \max\left\{\frac{n^+}{n^-}, \frac{n^-}{n^+}\right\}, $$ 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 $ \chi_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.