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Mathematics and Statistics Vol. 13(1), pp. 41 - 47
DOI: 10.13189/ms.2025.130105
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Reduced Second Zagreb Index and Bounds of Some Graph Operations

K. Rengalakshmi , S. Pethanachi Selvam ^*
PG and Research Department of Mathematics, The Standard Fireworks Rajaratnam College for Women, India

ABSTRACT

To link mathematics with the vast field of QSAR (quantitative structure-activity relationship) and QSPR (quantitative structure-property relationship) research, the idea of the concept of chemical graph theory is introduced. Topological indices refer to numerical values or descriptors that encode the structural properties of a molecular graph. There are numerous topological indices that have been created and applied as a tool in QSAR/QSPR research up to this point. Among those indices, the reduced second Zagreb index () has been established in recent times. Combining two graphs results in a new graph, like the lexicographic product of a cycle with n vertices with the path on two vertices results in a closed fence graph, and a path on n vertices with a path on two vertices results in a fence graph whose index can be easily computed by our obtained results. In this article, we compute the index for the join product, lexicographic product, and tensor product of any two simply connected graphs in terms of the first and second Zagreb index and the cardinality of the graphs' vertex and edge sets that are being used. For this, we use the degree of a vertex in the newly created graph that comes from an operation, as well as the vertex and edge set cardinality of the graphs involved in the process. In terms of maximum and minimum degree, we additionally establish certain lower and upper bounds for the aforementioned products. We further state the necessary and sufficient condition to obtain equality for the bounds. Furthermore, we deduce bounds on index for the earlier mentioned products of certain graphical structures, such as paths and cycles, and verify the index for a closed fence graph for application purposes. In this way, various operations can be performed to obtain different chemical structures that exist in our everyday lives. The structural and chemical characteristics of the obtained chemical structure attained by the graph invariant can be used in drug delivery, pharmaceutical research, and research purposes.

KEYWORDS
Graph Join, Lexicographic Product, Reduced Second Zagreb Index, Tensor Product

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] K. Rengalakshmi , S. Pethanachi Selvam , "Reduced Second Zagreb Index and Bounds of Some Graph Operations," Mathematics and Statistics, Vol. 13, No. 1, pp. 41 - 47, 2025. DOI: 10.13189/ms.2025.130105.

(b). APA Format:
K. Rengalakshmi , S. Pethanachi Selvam (2025). Reduced Second Zagreb Index and Bounds of Some Graph Operations. Mathematics and Statistics, 13(1), 41 - 47. DOI: 10.13189/ms.2025.130105.