A Spectral Lower Bound on Chromatic Numbers using p-Energy

Let AG be the adjacency matrix of a simple graph G, and let (G), f (G), q (G), (G) and f (G) denote its chromatic number, fractional chromatic number, quantum chromatic number, orthogonal rank and projective rank, respectively. For p 0, we define the positive and negative p -energies of G by Eₚ^+ (G) = 㶁 > ₀ ᵢᵖ, Eₚ^- (G) = 㶁 < ₀ |ᵢ|ᵖ, where ₁ ₙ are the eigenvalues of AG. We prove that for all p 0, (G) \f (G), q (G), (G) \ f (G) 1 + \ {Eₚ^+ (G) Eₚ^- (G), Eₚ^- (G) Eₚ^+ (G) \}^1{|p - 1|}. This result unifies and strengthens a series of existing bounds corresponding to the cases p \0, 2, \. In particular, the case p = 0 yields the inertia bound f (G) f (G) 1 + \n^+{n^-, n^-n^+\}, where n^+ and n^- denote the number of positive and negative eigenvalues of AG, 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 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.