Generalization of Randic ́ Index of the Non-commuting Graph for Some Finite Groups

Authors

  • Siti Rosllydia Dania Roslly College of Computing, Informatics and Mathematics, Universiti Teknologi MARA, Negeri Sembilan Branch, Seremban Campus, 70300 Seremban 3, Negeri Sembilan, Malaysia
  • Nur Fatimah Az Zahra Ab Halem College of Computing, Informatics and Mathematics, Universiti Teknologi MARA, Negeri Sembilan Branch, Seremban Campus, 70300 Seremban 3, Negeri Sembilan, Malaysia
  • Nur Syasya Sahira Zailani College of Computing, Informatics and Mathematics, Universiti Teknologi MARA, Negeri Sembilan Branch, Seremban Campus, 70300 Seremban 3, Negeri Sembilan, Malaysia
  • Nur Idayu Alimon College of Computing, Informatics and Mathematics, Universiti Teknologi MARA, Johor Branch, Pasir Gudang Campus, 81750 Masai, Johor, Malaysia
  • Siti Afiqah Mohammad College of Computing, Informatics and Mathematics, Universiti Teknologi MARA, Johor Branch, Segamat Campus, 85000 Segamat, Johor, Malaysia

DOI:

https://doi.org/10.11113/mjfas.v19n5.3047

Keywords:

Randić index, Non-commuting graph, Dihedral group, Generalized quaternion group, Quasi-dihedral groups

Abstract

Randić index is one of the classical graph-based molecular structure descriptors in the field of mathematical chemistry. The Randić index of a graph is calculated by summing the reciprocals of the square root of the product of the degrees of two adjacent vertices in the graph. Meanwhile, the non-commuting graph is the graph of vertex set whose vertices are non-central elements and two distinct vertices are joined by an edge if and only if they do not commute. In this paper, the general formula of the Randić index of the non-commuting graph associated to three types of finite groups are presented. The groups involved are the dihedral groups, the generalized quaternion groups, and the quasi-dihedral groups. Some examples of the Randić index of the non-commuting graph related to a certain order of these groups are also given based on the main results.

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Published

19-10-2023