bibkey: ghanbari2022somborenergy authors: Nima Ghanbari year: 2022 title: “On the Sombor characteristic polynomial and Sombor energy of a graph” doi: 10.1007/s40314-022-01957-5 url: https://arxiv.org/abs/2108.08552v1 claim: “There is no graph with integer-valued Sombor energy.” strata_touched:
- D5/S3/Combinatorics/Graph/SomborEnergyIntegerRefutation license: citation-only triage: anchor
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DOI: 10.1007/s40314-022-01957-5
Source: https://arxiv.org/abs/2108.08552v1
Conjecture 3.8, page 13 of the arXiv v1 PDF: “There is no graph with integer-valued Sombor energy.”
The Abstract, page 1, defines the Sombor matrix by the following sentence: “Let G be a simple graph with vertex set V(G) = {v₁, v₂, …, vₙ}. The Sombor matrix of G, denoted by A_SO(G), is defined as the n×n matrix whose (i,j)-entry is √(dᵢ²+dⱼ²) if vᵢ and vⱼ are adjacent and 0 for another cases.” It defines the energy as follows: “The Sombor energy En_SO of G is the sum of absolute values of the eigenvalues of A_SO(G).” Section 1 restricts graphs to finite simple graphs without directed, multiple or weighted edges or self-loops, and defines degree as the number of adjacent vertices.
The settling module uses this literal matrix and sums the absolute values of its Hermitian eigenvalues inline. Three copies of C₄ sharing one vertex give ten vertices, central degree six and all other degrees two. The module’s result refutes Conjecture 3.8 by proving that this graph has energy 48.