bibkey: alikhanighanbaridehghanizadeh2024ellipticsombor authors: Saeid Alikhani, Nima Ghanbari, Mohammad Ali Dehghanizadeh year: 2024 title: “Elliptic Sombor energy of a graph” doi: 10.48550/arXiv.2404.18622 url: https://arxiv.org/abs/2404.18622v1 claim: “There is no graph with integer-valued elliptic Sombor energy.” strata_touched:
- D5/S3/Combinatorics/Graph/EllipticSomborEnergyIntegerRefutation license: citation-only triage: anchor
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DOI: https://doi.org/10.48550/arXiv.2404.18622
Source: https://arxiv.org/abs/2404.18622v1
Conjecture 3.9, page 12: “There is no graph with integer-valued elliptic Sombor energy.”
The abstract, page 1, defines the matrix and energy:
Let G be a simple graph with vertex set V(G) = {v₁, v₂, …, vₙ}. The elliptic Sombor matrix of G, denoted by A_ESO(G), is defined as the n×n matrix whose (i,j)-entry is (dᵢ+dⱼ)√(dᵢ²+dⱼ²) if vᵢ and vⱼ are adjacent and 0 for another cases.
The elliptic Sombor energy E_ESO of G is the sum of absolute values of the eigenvalues of A_ESO(G).
Section 1, page 1, defines dᵢ as the number of vertices adjacent to vᵢ. The matrix is real and symmetric. Its characteristic-polynomial roots are counted with algebraic multiplicity when computing the energy.
Two copies of the four-cycle sharing one vertex give a seven-vertex graph with degrees 4, 2, 2, 2, 2, 2, 2. Its elliptic Sombor matrix has eigenvalues −56, −16, 0, 0, 0, 16, 56 and energy 144. This refutes Conjecture 3.9; it does not alter the paper’s separately established formulas for its listed graph classes.