The Sombor deletion correction term: extremal structure, exact coincidence classification, and the…

Abstract

The Sombor index SO(𝐺) = ∑𝑢𝑣∈𝐸 √𝑑2𝑢 + 𝑑2𝑣is 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 ∑𝑒 SO(𝐺 − 𝑒) = (𝑚 − 1) SO(𝐺) − 𝐶(𝐺) with a single correction term — an edge-additive index with an irrational kernel, which we call the Sombor deletion correction term 𝐶 . The paper develops the theory of 𝐶 along three axes. Structure (Sections 2–3): 𝐶 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 (𝑛,𝑚,𝑀1,𝑀2, 𝐹 , SO), the degree sequence, and small spectra. Exact algebra of the drops (Section 4): the per-edge drops Δ𝑒 = SO(𝐺) − SO(𝐺 − 𝑒) 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 — cross-type coincidences form exactly two families (based on √ 20 and √ 52), within-type coincidences are governed by a diagonal-union theorem and telescoping identities, and the all-degree classification is conjectural. Deck information (Sections 5–7): over all 288,266 graphs on 𝑛 ≤ 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); 𝐶 is injective modulo matching components on both ranges, but fails in general: an explicit collision with value 20 √ 2 − 4 √ 5 is constructed. On the 18 octane isomers 𝐶 separates 16 structures and raises a boiling-point regression from 𝑅2 = 0.559 (SO alone) to 0.814 (with 𝐶), an exploratory signal. All claims are proved, exhaustively verified, or machine-checked in Lean 4 (Section 8).

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BibTeX

@misc{gong2026sombor,
  title = {{The Sombor deletion correction term: extremal structure, exact coincidence classification, and the information content of edge decks, with applications to chemistry}},
  author = {Junyi Gong},
  howpublished = {ChemRxiv},
  year = {2026},
  month = sep,
  doi = {10.26434/chemrxiv.15009571/v1},
  url = {https://chemrxiv.org/doi/full/10.26434/chemrxiv.15009571/v1},
  note = {CC BY},
}

RIS

TY  - UNPB
TI  - The Sombor deletion correction term: extremal structure, exact coincidence classification, and the information content of edge decks, with applications to chemistry
AU  - Junyi Gong
PY  - 2026
DA  - 2026/09/28
PB  - ChemRxiv
DO  - 10.26434/chemrxiv.15009571/v1
UR  - https://chemrxiv.org/doi/full/10.26434/chemrxiv.15009571/v1
AB  - The Sombor index SO(𝐺) = ∑𝑢𝑣∈𝐸 √𝑑2𝑢 + 𝑑2𝑣is 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 ∑𝑒 SO(𝐺 − 𝑒) = (𝑚 − 1) SO(𝐺) − 𝐶(𝐺) with a single correction term — an edge-additive index with an irrational kernel, which we call the Sombor deletion correction term 𝐶 . The paper develops the theory of 𝐶 along three axes. Structure (Sections 2–3): 𝐶 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 (𝑛,𝑚,𝑀1,𝑀2, 𝐹 , SO), the degree sequence, and small spectra. Exact algebra of the drops (Section 4): the per-edge drops Δ𝑒 = SO(𝐺) − SO(𝐺 − 𝑒) 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 — cross-type coincidences form exactly two families (based on √ 20 and √ 52), within-type coincidences are governed by a diagonal-union theorem and telescoping identities, and the all-degree classification is conjectural. Deck information (Sections 5–7): over all 288,266 graphs on 𝑛 ≤ 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); 𝐶 is injective modulo matching components on both ranges, but fails in general: an explicit collision with value 20 √ 2 − 4 √ 5 is constructed. On the 18 octane isomers 𝐶 separates 16 structures and raises a boiling-point regression from 𝑅2 = 0.559 (SO alone) to 0.814 (with 𝐶), an exploratory signal. All claims are proved, exhaustively verified, or machine-checked in Lean 4 (Section 8).
ER  -

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