Proof of an open problem on the Sombor index

The Sombor index is one of the geometry-based descriptors, which was defined as S O ( G ) = ∑ u v ∈ E ( G ) d u 2 + d v 2 , where d u (resp. d v ) denotes the degree of vertex u (resp. v ) in G . In this note, we determine the graphs among the set of graphs with vertex connectivity (resp. edge conne...

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Bibliographic Details
Published inJournal of applied mathematics & computing Vol. 69; no. 3; pp. 2465 - 2471
Main Author Liu, Hechao
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2023
Springer Nature B.V
Subjects
Apexes
Applied mathematics
Computational Mathematics and Numerical Analysis
Graphs
Mathematical and Computational Engineering
Mathematics
Mathematics and Statistics
Mathematics of Computing
Original Research
Theory of Computation
05C07
Edge connectivity
05C09
05C92
Sombor index
Vertex connectivity
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Summary:The Sombor index is one of the geometry-based descriptors, which was defined as S O ( G ) = ∑ u v ∈ E ( G ) d u 2 + d v 2 , where d u (resp. d v ) denotes the degree of vertex u (resp. v ) in G . In this note, we determine the graphs among the set of graphs with vertex connectivity (resp. edge connectivity) at most k having the maximum and minimum Sombor indices, which solves an open problem on the Sombor index proposed by Hayat and Rehman [On Sombor index of graphs with a given number of cut-vertices, MATCH Commun. Math. Comput. Chem. 89 (2023) 437–450]. For some conclusions of the above paper, we first give some counterexamples, then provide another simple proof about the minimum Sombor indices of graphs with n vertices, k cut vertices and at least one cycle.
ISSN:1598-5865
1865-2085
DOI:10.1007/s12190-023-01843-1