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 $p\ge 0$, we define the positive and negative $p$-energies of $G$ by $${\mathcal{E}}_p^{+}(G)=\sum _{{\lambda }_i>0}{\lambda }_i^p,{\mathcal{E}}_p^{-}(G)=\sum _{{\lambda }_i<0}\mathrm{\mid }{\lambda }_i{\mathrm{\mid }}^p,$$ where ${\lambda }_1\ge \cdots \ge {\lambda }_n$ are the eigenvalues of $A_G$. We prove that for all $p\ge 0$, $$\chi (G)\ge \{{\chi }_f(G),{\chi }_q(G),\xi (G)\}\ge {\xi }_f(G)\ge 1+\max {\{\frac{{\mathcal{E}}_p^{+}(G)}{{\mathcal{E}}_p^{-}(G)},\frac{{\mathcal{E}}_p^{-}(G)}{{\mathcal{E}}_p^{+}(G)}\}}^{\frac{1}{\mathrm{\mid }p-1\mathrm{\mid }}}\mathrm{.}$$ This result unifies and strengthens a series of existing bounds corresponding to the cases $p\in \{0,2,\mathrm{\infty }\}$. In particular, the case $p=0$ yields the inertia bound $${\chi }_f(G)\ge {\xi }_f(G)\ge 1+\max \{\frac{n^{+}}{n^{-}},\frac{n^{-}}{n^{+}}\},$$ 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.