The Sombor deletion correction term: extremal structure, exact coincidence classification, and the information content of edge decks, with applications to chemistry
Year: 2026
Type: preprint
Abstract: The Sombor index SO(πΊ) = βπ’π£βπΈ βπ2π’ + π2π£is a distinguished member of the family of edge-additive degree-based topological indices. We study what happens to SO when an edge is deleted. Summed over all edges, the deletions obey the exact identity βπ SO(πΊ β π) = (π β 1) SO(πΊ) β πΆ(πΊ) with a single correction term β an edge-additive index with an irrational kernel, which we call the Sombor deletion correction term πΆ . The paper develops the theory of πΆ along three axes. Structure (Sections 2β3): πΆ is a linear functional of the edge-type counts, with sharp two-sided bounds against Zagreb-type polynomials and a star-maximisation theorem with full equality characterisation, closed forms for regular graphs and products, and non-expressibility certificates against (π,π,π1,π2, πΉ , SO), the degree sequence, and small spectra. Exact algebra of the drops (Section 4): the per-edge drops Ξπ = SO(πΊ) β SO(πΊ β π) lie in a fixed multiquadratic field, a normal-form procedure decides their equalities exactly, and an exhaustive computation classifies all coincidences up to degree bound 8 β cross-type coincidences form exactly two families (based on β 20 and β 52), within-type coincidences are governed by a diagonal-union theorem and telescoping identities, and the all-degree classification is conjectural. Deck information (Sections 5β7): over all 288,266 graphs on π β€ 9 vertices and all 89,402 chemical graphs on ten vertices, the Sombor deck and the edge-type deck induce the same partition (53,984 classes); πΆ is injective modulo matching components on both ranges, but fails in general: an explicit collision with value 20 β 2 β 4 β 5 is constructed. On the 18 octane isomers πΆ separates 16 structures and raises a boiling-point regression from π
2 = 0.559 (SO alone) to 0.814 (with πΆ), an exploratory signal. All claims are proved, exhaustively verified, or machine-checked in Lean 4 (Section 8).
Source: ChemRxiv
Author: Junyi Gong
Institution: Shenzhen MSU-BIT University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214655617