trureturing — Discovering truth

bibkey: das2026sgbcyclic
authors: Shrabani Das; Ahmad Erfanian; Rajat Kanti Nath
year: 2026
title: "Certain topological indices and spectral properties of SGB-graphs of finite cyclic groups"
doi: 10.48550/arXiv.2602.07587
url: https://arxiv.org/abs/2602.07587v1
claim: "Section 5 closing extension of the complete (a)-(c) list to any cyclic group."
strata_touched:
  - D5/S3/Combinatorics/Graph/SubgroupGenerating/CyclicExtensionRefutation
license: citation-only
triage: anchor

Verified locator

DOI: 10.48550/arXiv.2602.07587

Source: https://arxiv.org/abs/2602.07587v1

The fixed source is arXiv:2602.07587v1, by Shrabani Das, Ahmad Erfanian and Rajat Kanti Nath. Section 1 defines the subgroup-generating bipartite graph of a finite group. Its two vertex parts are all ordered pairs of group elements and all subgroups, with a cross edge precisely when the pair generates the subgroup. Section 2 explicitly uses disjoint union.

Section 5 defines adjacency, degree, Laplacian, signless Laplacian and common-neighbour matrices, real spectra with multiplicities, and their absolute energies. Only Laplacian and signless Laplacian energies subtract the average degree of the entire graph. Hyperenergeticity compares each energy with that of the complete graph having the same number of vertices.

The final paragraph summarizes ten assertions for the known orders $p^n$, $pq$, $p^{2}q$ and $p^2{q}^2$, with distinct primes $p,q$ and $n\ge 1$: adjacency nonintegrality; L, Q and CN integrality; strict hypoenergeticity; four non-hyperenergeticity assertions; and $E\le LE$. It then invites extension to any cyclic group. This is an unlabelled extension invitation, not a numbered conjecture. The finite context includes positive order one; the prime-power hypothesis $n\ge 1$ does not impose a nontriviality convention on that invitation. The known families belong to Das–Erfanian–Nath: Theorem 2.1 recalls their earlier work, and Theorems 2.2–2.4 and 5.3–5.18 supply the families used in the summary.

The Lean result negates the entire finite-cyclic universal, using the actual graph of the order-one group and its integral adjacency spectrum. It asserts nothing about private author intent or an extension restricted to nontrivial groups.

The inspected original displays CC BY-NC-SA 4.0. This note supplies bibliographic attribution and independently written mathematical description; it imports no paper program, figure or expressive proof text. The source version and bounded prior-status assessment are recorded in the corresponding Problems dossier.