Topological Indices of the Non-commuting Graph for Generalised Quaternion Group
- PDF / 277,505 Bytes
- 7 Pages / 439.37 x 666.142 pts Page_size
- 49 Downloads / 177 Views
Topological Indices of the Non-commuting Graph for Generalised Quaternion Group Nor Haniza Sarmin1
· Nur Idayu Alimon1 · Ahmad Erfanian2
Received: 1 November 2018 / Revised: 14 May 2019 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019
Abstract A topological index is a numerical value associated with chemical constitution for correlation of chemical structure with various physical properties and chemical reactivity. It is calculated from a graph representing a molecule. Meanwhile, the non-commuting graph, G of G, is defined as a 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. The main objective of this article is to determine the general formula of some topological indices, namely Wiener index, first Zagreb index and second Zagreb index for the non-commuting graph associated with generalised quaternion group in terms of n. Keywords Wiener index · Zagreb index · Non-commuting graph · Generalised quaternion group Mathematics Subject Classification 05C12
1 Introduction Topological indices have become increasingly important in the prediction of physical properties in chemistry and biology. They can also be used as simple descriptors
Communicated by V. Ravichandran.
B
Nor Haniza Sarmin [email protected] Nur Idayu Alimon [email protected]
1
Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 Johor Bahru, Johor, Malaysia
2
Department of Pure Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran
123
N. H. Sarmin et al.
in a comparison with physical, chemical or biological parameters of molecules in quantitative structure property relationship (QSPR) and in quantitative structure active relationship (QSAR). A topological representation of a molecule is called molecular graph where the atoms represent the vertex set and the covalent bonds represent the edge set. One of the most widely known topological indices is the Wiener index which has been introduced by Wiener in 1947 [1]. It has been used to predict the boiling point of paraffin. Meanwhile, the Zagreb index has been used since 1972 which it was first introduced by Trinajsti´c [2]. The idea of the Zagreb index is found when they examined the dependence of total π -electron energy on molecular structure. The presentation of the generalised quaternion group, Q 4n , is given in the following: Q 4n = a n = b2 , a 2n = b4 = 1, b−1 ab = a −1 , where n ≥ 2. This paper consists of three sections. Section 1 is the introduction section, followed by Sect. 2, namely the preliminaries where some basic concepts, definitions and previous results on group theory and graph theory are stated. In Sect. 3, the main results which are the new theoretical results on the generalisation of Wiener index and Zagreb index of the non-commuting graph associated with generalised quaternion group are presented.
2 Preliminaries In this section, some basic concepts, definitio
Data Loading...
