Minimal Trees for the Euler Sombor Index Revisited: A Correction and the First Thirteen Minima

Abstract

Correction and extension of Theorems 3 and 4 of B. Khanra and S. Das, "Euler Sombor index of trees, unicyclic and chemical graphs", MATCH Commun. Math. Comput. Chem. 94 (2025) 525–548 (doi:10.46793/match.94-2.525K). Parts (c) and (d) of Theorem 3 of that paper are incorrect: the proof gives the edge joining two adjacent degree-3 vertices the weight 2√3 of a (2,2)-edge instead of √27, so the printed fourth-minimum value 4√7 + 4√19 − 18√3 + 2√3·n is not the Euler Sombor index of any tree, the family presented as fifth-minimal is in fact fourth-minimal for every n ≥ 7, and the corrected value 4√7 + 4√19 − 17√3 + 2√3·n is fifth-minimal. Via a thread decomposition of the index the manuscript classifies the first five minima for all n ≥ 7 and extends the classification to the first thirteen minima for all n ≥ 11 (nine explicitly ordered families of subdivided double stars S(3,3)). A boundary error in Theorem 4(iii) (unicyclic graphs, n ≥ 4 corrected to n ≥ 5) is also fixed. Contents: the manuscript (paper.pdf), a self-contained exact verification script (verify_euler_sombor.py; all Euler Sombor values handled as integer vectors over square roots of squarefree integers, ordering certified by integer separation inequalities, no floating-point tolerances) covering all 204,990 non-isomorphic trees with 7 ≤ n ≤ 18 and all 1,039 unicyclic graphs with 4 ≤ n ≤ 10, its full output (verify_output.txt, all checks pass), and a README.