Steiner 3-Wiener Index of Zigzag Polyhex Nanotubes.

Background: Let G be a connected graph and S be a k element subset of the vertex set V(G) of G. Steiner distance is a natural generalization of the usual graph distance. The Steiner-k distance dG(S) between the vertices of S is the minimum size among all connected subgraphs whose vertex set contains S.

M, NM-polynomials Based on Reverse, Reduced Reverse Degree and Neighborhood Degree Based Topological Indices with Applications to Bond Energy of Y-Junction Nanotubes.

Background: In graph theory, M polynomials like the matching polynomial are very crucial in examining the matching structures within graphs, while NM polynomials extends this to analyze non-matching edges. These polynomials are important in many fields, including chemistry and network architecture. They support the derivation of topological indices for protein structure analysis, network communication optimization, and drug design in QSAR/QSPR investigations.

Novel Decision Modeling for Manufacturing Sustainability under Single-Valued Neutrosophic Hesitant Fuzzy Rough Aggregation Information.

We developed a multicriteria decision-making method based on the list of novel single-valued neutrosophic hesitant fuzzy rough (SV-NHFR) weighted averaging and geometric aggregation operators to address the uncertainty and achieve the sustainability of the manufacturing business. In addition, a case study on choosing the optimum elements for a sustainable manufacturing sector was carried out. The proposed decision support method is then compared to other relevant methodologies, and a validity test is performed to show the reliability and validity of the new methodology.