A New Alternative to Szeged, Mostar, and PI Polynomials—The SMP Polynomials

Martin Knor and Niko Tratnik ()
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Martin Knor: Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 810 05 Bratislava, Slovakia
Niko Tratnik: Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška Cesta 160, 2000 Maribor, Slovenia

Mathematics, 2023, vol. 11, issue 4, 1-15

Abstract: Szeged-like topological indices are well-studied distance-based molecular descriptors, which include, for example, the (edge-)Szeged index, the (edge-)Mostar index, and the (vertex-)PI index. For these indices, the corresponding polynomials were also defined, i.e., the (edge-)Szeged polynomial, the Mostar polynomial, the PI polynomial, etc. It is well known that, by evaluating the first derivative of such a polynomial at x = 1 , we obtain the related topological index. The aim of this paper is to introduce and investigate a new graph polynomial of two variables, which is called the SMP polynomial, such that all three vertex versions of the above-mentioned indices can be easily calculated using this polynomial. Moreover, we also define the edge-SMP polynomial, which is the edge version of the SMP polynomial. Various properties of the new polynomials are studied on some basic families of graphs, extremal problems are considered, and several open problems are stated. Then, we focus on the Cartesian product, and we show how the (edge-)SMP polynomial of the Cartesian product of n graphs can be calculated using the (weighted) SMP polynomials of its factors.

Keywords: SMP polynomial; edge-SMP polynomial; Cartesian product; Szeged index; Mostar index; PI index (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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