bibkey: movahedi2025diminishedsombor authors: F. Movahedi year: 2025 title: “Diminished Sombor matrix, spectral radius, and energy of the graphs” doi: 10.48550/arXiv.2508.06531 url: https://arxiv.org/abs/2508.06531v1 claim: “There does not exist a graph whose diminished Sombor energy is an integer value.” strata_touched:

• D5/S3/Combinatorics/Graph/DiminishedSomborEnergyRefutation license: citation-only triage: anchor

Diminished Sombor energy

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DOI: 10.48550/arXiv.2508.06531

Source: https://arxiv.org/abs/2508.06531v1

The matrix and energy definitions are on page 2. Conjecture 5.1 is in Section 5, page 19.

Definitions

Page 2: “Motivated by this newly introduced index, and following on the approach in [5, 28], we introduce the diminished Sombor matrix for the graph G, denoted by ℳ = M_DS(G) = (μ_ij), of order n as follows”

For a finite simple graph with vertex degrees $d_i$,

$${\mu }_{ij}=\{\begin{array}{ll}\frac{\sqrt{d_i^2+d_j^2}}{d_i+d_j}& \text{if }{v}_i{v}_j\in E,\\ 0& \text{otherwise}.\end{array}$$

Page 2: “We define the diminished Sombor energy as follows”

$$E_{DSO}(G)=\sum _{i=1}^n|{\lambda }_i|,$$

where the ${\lambda }_i$ are the real eigenvalues of $M_{DS}(G)$, counted with multiplicity.

Conjecture 5.1

Section 5, page 19: “There does not exist a graph whose diminished Sombor energy is an integer value.”

The graph domain includes edgeless graphs: Corollary 4.3, page 17, lists $\overline{K_n}$ as an equality case in the upper energy bound. On an edgeless graph the non-edge branch makes every matrix entry zero. Its eigenvalues and energy are therefore zero, an integer. This refutes the literal conjecture; the restriction to graphs with at least one edge remains open.