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Journal of Discrete Mathematical Sciences and Cryptography

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Journal of Discrete Mathematical Sciences and Cryptography
Open Access Research Article

Relationship between the inverse sum indeg index of graph operations and their factors

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pp. 1181–1199Vol. 29Issue 3March 2026DOI: 10.47974/JDMSC-2193 Crossmark XML
Received:
06 Mar 2024
Published Online:
22 Dec 2025
Article type:
Research Article
Language:
EN
Article no.:
JDMSC-2193
Pages:
1181–1199

Abstract

The inverse sum indeg (ISI) index of a simple graph 𝒦 is expressed by  ISI(𝒦) = ∑ς𝒦 ς’𝒦∈E𝒦 d𝒦  (ς𝒦)d𝒦 (ς’𝒦)/d𝒦 (ς𝒦) + d𝒦 (ς’𝒦).   in which E𝒦 represents the edge set of the graph 𝒦 and d𝒦 (ς𝒦) indicates the degree of the vertex ς𝒦 in 𝒦. The ISI index is one of the 20 selected indices from a group of 148 discrete Adriatic indices that have been proposed as remarkable predictors of physico-chemical properties of molecular structures. In addition, this index has been detected as an excellent measure for predicting the total surface area of isomers of octane. In this study, we establish a comparison between the ISI index for various standard graph operations, including the sum, corona product, disjunctive product, Cartesian product, lexicographic product, and strong product, alongside the ISI index of their respective factors. This comparison is achieved by presenting new and sharp upper and lower bounds on the ISI index for the aforementioned graph operations, expressed based on the ISI indices of their respective factors. The necessary and sufficient conditions for the inequalities to hold as equalities are also analyzed. In the specific case where all factors are regular graphs, we provide exact formulas for the ISI index. Utilizing these results, we present precise values for the ISI index of various graphs of mathematical or chemical significance, including cone graphs, windmill graphs, thorn graphs, C4-nanotori, Rook’s graphs, prism graphs, and closed fence graphs. 

Keywords

Vertex degreeGraph invariantGraph operations

Subject Classifications

05C0905C0705C76

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