bibkey: albalahidasalibarmanhamza2025hyperbolicsombor authors: A. M. Albalahi, S. Das, A. Ali, J. Barman, A. E. Hamza year: 2025 title: “On the hyperbolic Sombor index and its counterpart” doi: 10.47443/dml.2025.176 url: https://www.dmlett.com/archive/v16/DML25_v16_pp108-115.pdf claim: “A graph minimizing (maximizing, respectively) the CDSO index (HSO index, respectively) among fixed-order connected graphs with cyclomatic number ℓ(≥ 1) has a vertex adjacent to all other vertices.” strata_touched:

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

The complementary diminished Sombor index

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DOI: 10.47443/dml.2025.176

Source: https://www.dmlett.com/archive/v16/DML25_v16_pp108-115.pdf

Discrete Mathematics Letters 16, pages 108–115. The preprint is arXiv:2510.24809v1.

Definitions

Page 109: “We drop the factor 1/√2 from the expression (1) and call the resulting formula the complementary diminished Sombor (CDSO) index and denote it by ᶜDSO. Hence, for a graph G, we have”

$${}^c\mathrm{DSO}(G)=\sum _{uv\in E(G)}\frac{\sqrt{(d(u){)}^2+(d(v){)}^2}}{\max \{d(u),d(v)\}}.$$

Each undirected edge occurs once; the graphs are simple and finite.

Page 115: “We recall that trees can be considered connected graphs of cyclomatic number 0, where the cyclomatic number of a graph is the minimum number of edges whose removal makes the graph acyclic.”

Conjecture 4.1

Page 115: “A graph minimizing (maximizing, respectively) the CDSO index (HSO index, respectively) among fixed-order connected graphs with cyclomatic number ℓ(≥ 1) has a vertex adjacent to all other vertices.”

The CDSO half quantifies over every minimizer at each fixed order and positive cyclomatic number. The HSO half concerns maximizers and is a separate assertion.