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Volume 2021 |Article ID 9913529 | https://doi.org/10.1155/2021/9913529

Tanweer Ul Islam, Zeeshan Saleem Mufti, Aqsa Ameen, Muhammad Nauman Aslam, Ali Tabraiz, "On Certain Aspects of Topological Indices", Journal of Mathematics, vol. 2021, Article ID 9913529, 20 pages, 2021. https://doi.org/10.1155/2021/9913529

On Certain Aspects of Topological Indices

Academic Editor: Riaz Ahmad
Received04 Mar 2021
Revised27 Mar 2021
Accepted20 Apr 2021
Published03 May 2021

Abstract

A topological index, also known as connectivity index, is a molecular structure descriptor calculated from a molecular graph of a chemical compound which characterizes its topology. Various topological indices are categorized based on their degree, distance, and spectrum. In this study, we calculated and analyzed the degree-based topological indices such as first general Zagreb index , geometric arithmetic index , harmonic index , general version of harmonic index , sum connectivity index , general sum connectivity index , forgotten topological index , and many more for the Robertson apex graph. Additionally, we calculated the newly developed topological indices such as the and Sanskruti index for the Robertson apex graph G.

1. Introduction

The Robertson apex graph (see Figure 1) was introduced by Neil Robertson. Robertson’s apex graph is the graph which has 15-vertex set and total number of edges is 32. In this graph, each vertex has the degree 4 except one vertex which has degree 6. The Robertson apex graph is formed by adding an additional apex vertex in the (planar) rhombic dodecahedral graph which is associated to all degree 3 vertex or by contracting two opposite vertices of the tesseract graph. The apex graph discussed here is in the form of unit, where “” represents the number of rows and “” represents the number of columns. First, we have found the degree-based topological indices of unit, unit, and so on, and finally, we have generated the formulas for the degree-based topological indices of unit.

A graph is a planar graph which can be drawn without any edge crossing. In an apex graph, the removal of a vertex gives a planar graph. Sometimes an apex graph is also called a virtually planar graph and the apices of the graph are the set of vertices whose removal reduces the graph planar. Apex graph differs from the critical nonplanar graph in which an apex graph requires only that their exists at least one vertex whose removal gives a planar graph although a critical nonplanar graph requires that removal of every vertex make a planar graph.

A graph can be recognized by a polynomial, a matrix, a sequence of numbers, or a numeric number which represents the whole graph, and these representations are designed to be uniquely defined for that graph [14]. A topological index, which is a graph invariant that does not depend on the classification or graphic demonstration of the graph, is a numerical parameter mathematically derived from the graph. The topological indices of molecular graphs are generally used for establishing correlations between the arrangement of a molecular compound and its physico-chemical properties or biological activity. Topological indices are major tools for analyzing many physico-chemical properties of molecules without performing any testing. Some most significant types of topological indices of graphs are distance-based topological indices, degree-based topological indices, and spectrum-based topological indices. One of the most investigated categories of topological indices used in mathematical chemistry is called degree-based topological indices, which are defined in terms of the degrees of the vertices of a graph. Thus, we can write the definition of such a topological index in the form given aswhere G = (V(G), E(G)) is a simple, undirected, and connected graph and d(u) denotes the degree of the vertex u. Topological indices also called molecular descriptors are presented to explain the physiochemical properties of molecules [57]. For more details on topological indices, see [812].

Gutman and Das proposed the first general Zagreb index in [13]:

Yuan et al. in 2009 defined geometric-arithmetic (GA) index and compared GA index by the well-known Randic index [14]:

Zhong [15] introduced the harmonic index which was described by

Yan et al. [16] introduced the general version of harmonic index which was defined by

In 2009, Zhou et al. [17] defined the sum-connectivity index which was formulated by

Few years ago, Yan et al. [16] introduced the general sum connectivity index as follows:

In 2015, Gao et al. [18] reinvestigated the forgotten topological index or (F-index) and it is denoted by F(G):

Arithmetic-geometric index [19] is defined as

Multiplicative geometric arithmetic index [20] is stated as follows:

Multiplicative sum connectivity index [21] is defined as

The index [19] is defined as

In 2016, Hosamani [22] formulated the Sanskruti index S(G) of a general graph G. It is defined as

2. Results and Discussion

Theorem 1. Let be a graph with and . Then,

Proof. Let be a graph and r is a real number. Then, by using Table 1, we have


Degree of vertexNo. of vertices

46mn + 18n + 18m + 44
65mn + 5n + 5m + 5
82mn + 4n + 3m + 6
92m + 2
10mn + n

Total40mn + 72m + 72n + 136

Theorem 2. Let be a graph with edges and vertices.(1)(2)(3)(4)(5)(6)(7)

Proof. (1) The geometric aritheoremetic index is defined asThis implies thatBy using Table 2, we getAfter simplification, we obtain(2) The aritheoremetic-geometric index is defined asThis implies thatBy using Table 2, we obtainAfter simplification, we obtain(3) The harmonic index is defined asThis implies thatBy using Table 2, we obtainAfter simplification, we obtainHence, it is proved.(4) The general harmonic index is defined asThis implies thatBy using Table 2, we obtainAfter simplification, we obtainHence, it is proved.(5) The sum-connectivity index is defined asThis implies thatBy using Table 2, we obtainAfter simplification, we obtainwhich completes the proof.(6) The general sum-connectivity index is defined asThis implies thatBy using Table 2, we obtainAfter simplification, we obtainHence, it is proved.(7) The F-index is defined asThis implies thatBy using Table 2, we obtainAfter simplification, we obtain


No. of edges

4mn + 20n + 20m + 60
4mn + 4n + 4m + 4
8mn + 24n + 16m + 40
12m + 12
4mn + 4n
6mn + 6m + 6n + 6
8mn + 8n + 8m + 8
6m + 6
6mn + 6n

Total40mn + 72m + 72n + 136

Theorem 3. Let G be a graph with edges and vertices.(1)(2)

Proof. (1) Multiplicative geometric aritheoremetic index is defined asThis implies thatBy using Table 2, we obtainAfter simplification, we obtained that(2) Multiplicative sum connectivity index is defined asThis implies that