Structural analysis of the signless Brouwer conjecture: projection thresholds, extremal families, and the Laplacian–signless gap

Abstract

We study the signless Brouwer conjecture (Ashraf–Omidi–Tayfeh-Rezaie): the inequality S_k(Q) ≤ m + C(k+1,2) for the sum of the k largest eigenvalues of the signless Laplacian Q = D + A of a graph with m edges, where C(k+1,2) denotes the binomial coefficient (k+1)(k)/2. We develop a structural framework centred on six results. A KKT threshold theorem (all k) characterises extremal graphs as spectral threshold graphs of rank k. We give a conjectural extremal family with a closed-form equitable quotient, and exact closed-form values of the k=3 maximum for n ≤ 6. A projection reformulation expresses the conjecture at level k as V(I_k) ≤ C(k+1,2) for a convex function V on the positive-semidefinite cone. Our main quantitative result is an explicit bound S_3(Q) ≤ m + 6.360043 for every graph, improving Lew's previous m + 3 + 10·sqrt(3) ≈ m + 20.32. It rests on two ingredients, both verified in exact rational arithmetic: a two-point Delsarte certificate giving S_3(Q) ≤ m + 7, and a realisability constraint invisible to the two-point relaxation — the Gram matrix of a tight frame is a rank-3 projection, so G ≼ I (not merely G ≽ 0), a constraint whose off-diagonal content is intrinsically three-point. A single rank-one realisability multiplier is worth more than the entire Schoenberg hierarchy, which we show saturates completely. We prove a completeness theorem for this certificate family: on the working grid its optimal constant lies in the interval [6.347176425, 6.360043] (both endpoints certified exactly), independent of the rank of the multiplier and of the degree of any adjoined Schoenberg block, with the obstruction identified as a degenerate dual plateau. Finally, we explain why the bound C(k+1,2) resists proof through four independent obstructions, all tracing to a single geometric phenomenon: the extremal configurations are a blow-up limit in which a vanishing dust carries a constant share of the excess. The conjecture at k=3 remains open and provably beyond the reach of this certificate family.