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.