The Extremal Moment Gap Between the Annihilation Number and the First Zagreb Index

For a finite simple graph G with n vertices, m ≥ 1 edges, degree sequence d₁ ≤ ··· ≤ dₙ, annihilation number a(G) = max{k : ∑_{i≤k} dᵢ ≤ m} and first Zagreb index M₁(G) = ∑ᵥ d(v)², put B(G) = (n/2)(1 + √(1 − 4m²/(nM₁(G)))) and Γ(G) = ⌊B(G)⌋ − a(G). We prove the universal inequality a(G) ≤ ⌊B(G)⌋, so that Γ ≥ 0, by applying the Cauchy–Schwarz inequality separately to the head and the tail of the degree sequence. An exact head/tail decomposition of M₁ and an exact normalization by the degree variance reduce the extremal problem for Γ, without any a priori assumption on the scale of the degrees, to the maximization of an explicit strictly concave function of one real variable. Solving that problem yields, for every n-vertex graph, Γ(G) ≤ n/2 − (3/4)n^(2/3) − (5/16)n^(1/3) + O(1). A connected construction (one universal vertex joined to both endpoints of each of p vertex-disjoint paths covering the remaining vertices, with degree sequence (2^(n−1), 2p)) attains this value up to an additive constant. Hence the unrestricted and the connected extremal functions Γmax(n) and Γconn(n) each equal n/2 − (3/4)n^(2/3) − (5/16)n^(1/3) + O(1), and in particular differ by O(1). To the best of our knowledge, the moment-gap invariant Γ and this asymptotic expansion do not appear in the literature we located. Terminology note: This work introduces the term “Thokal moment-gap formula” for the invariant defined in Definition 3.2, ΓT(G) = ⌊B(G)⌋ − a(G). The terminology is introduced by the present work and is not asserted as an established name in the literature. This manuscript is self-contained: every result is proved in the paper without dependence on computational verification. Computational experiments are included only as reproducibility and sanity checks for the derived inequalities and the connected construction. These checks include all 1,245 nonempty graphs in the NetworkX graph atlas with n ≤ 7, 5,000 deterministic Erdős–Rényi G(n,p) trials with n sampled uniformly from 8 to 80 and p uniformly from 0.05 to 0.95, and validation of the connected construction for feasible instances with n = 10,…,200. The accompanying public repository contains the proof-related documentation, validation scripts, and GitHub Actions workflows used for reproducibility:https://github.com/aadityat23/Thokal-Moment-Gap This work is deposited as Version 1.0. The manuscript has not undergone external peer review.