Web of Science (Emerging Sources Citation Index), Scopus, ISC

Document Type : Original Research Article


1 School of Mathematics, Minhaj University, Lahore, Pakistan

2 Faculty of Education, Van Yuzuncu Yıl University, Zeve Campus, Tuşba, 65080, Van, Turkey


Topological indices are extremely useful for analyzing various physical and chemical properties associated with a chemical compound. A topological index describes molecular structures by converting them into certain real numbers. Topological indices are used in the development of quantitative structure-activity relationships (QSARs) in which the biological activity of molecule correlated with their chemical structure. The chemical shape of benzene molecule is very common in nano-science, chemistry, and physics. The circumcoronene collection of benzenoid  generates from the benzene molecules. Jahangir graph  is a generalized wheel graph that consists of  circular vertices and a center vertex connected to every 𝑚𝑡 vertex on the circle. In this article, we will compute the topological indices of the middle graph of the circumcoronene series of benzenoids  and Jahangir graph . In addition, comparison of the middle graph of the circumcoronene series of benzenoid  and Jahangir graph  are presented numerically and graphically.

Graphical Abstract

Study of middle graph for certain classes of graph by applying degree-based topological indices


Main Subjects


Consider a molecular graph G=(V,E), such a graph with vertex set  V(G) indicates the atoms and edge set E(G) indicates chemical bonds. A degree is represented by dө {өᵋV(G)} which is defined as the number of edges incident with ө (For unspecified terminologies and more details [1]).

Graph theory is branch of mathematics that has been applied in virtually every field of study. The usage of topological indices in QSPR/QSAR studies has taken important concentration in recent years. Graph theory is used to assess the linkage among several topological indices of certain graphs that generated by some graph operations that are middle graph, total graph, semi-total graph, and the strong double graph etc. Topological indices are numerical parameters of a graph molecule that characterize its topology [2]. The first topological index to be applied in chemistry is the Wiener index. To be more precisely, Harold Wiener introduced this intriguing index in 1947 to assess the physical characteristics of the type of alkane known as paraffin [3].

The symmetric division degree index (SD) of connected graph (G) [4] is defined as follows:

Where,  dө and dᵚ are the degrees of vertex  ө and ᵚ in G.

The sum-Connectivity index [5] is defined as follows:

Randic connectivity index is widely used in mathematical chemistry, due to its wide applications in both mathematics and chemistry. It is defined [6] in the following equation:

For more wide-ranging and comprehensive details, we offer the readers to follow the following articles [9-13, 17-40].

Definition 1.1. A graph that contains a cycle  Cmt having an extra vertex which is adjacent to t vertices of Cmt at the distance  to each other on the Cmt. In Jahngir graph [14] (Jm,t), where t2 and m3. The number of vertices and edges is  mt+1 and  mt+m respectively. Jahangir graphs J(3,2), J(3,3), and J(3,t) are displayed in Figure 1.

Definition 1.2. Circumcoronene series of benzenoid  (Hs) where, (s≥1) is one family that is generated from benzene  C6 on circumference [15]. The number of vertices are 6s2 and edges are 9s2-3s ,  in this series of benzenoid. The Circumcoronene series of benzenoids are designated in Figure 2.

Definition 1.3. The middle graph [16] of any graph G is attained by adding a new vertex to each of its edge and connecting by edges any pairs of those new vertices which lie on the adjacent edges of the graph. The middle graph of graph  G is represented by  M(G) For example, the middle graph of the Jahangir graph  (J(a,a)) is depicted in Figure 3.

Result for the middle graph of circumcoronene series of benzenoid graph  

In this section, we calculate the degree-based indices of the middle graph of (Hs), where s≥2.

Theorem 4.1. Let [M(Hs)] be the middle graph of circumcoronene series of benzenoid. Then,

Proof: The middle graph of circumcoronene series of benzenoid M(Hs) where s≥2,  has  6s vertices of degree 2,  6s(s-1) vertices of degree 3, 6 vertices of degree 4 , 6s vertices of degree  5 and 9s2-9s-6 vertices of degree 6.

In M(Hs), we get edge of type E(2,4), E(2,5), E(3,5), E(3,6), E(4,5), E(5,5), E(5,6), and E(6,6). Table 3 lists the number of edges.

Now by using Table 3 and the Equation (1), we obtain the desired results, i.e.,


In this section, we provide the comparison of the above-computed topological indices numerically and the graphically. The numerical comparison of M(Hs) where s=2, 3, ..., 10,  as presented in Table 4, and the graphical comparison is displayed in the Figure 6.


Topological indices help to understand the information about biological activity, chemical reactivity, and physical characteristics of chemical compounds. We derived the general formulas of some of the topological indices based on the degree of vertex i.e. sum connectivity index SC, Randic connectivity index (RC), Symmetric division degree index SD, Harmonic index (H), the first Zagreb index M1 and the second Zagreb index  M2 of the middle graph of Jahangir graph J(3,t). These outcomes can be employed to further understand the topological characteristics of graphs. The comparison of attained analytical expressions is expressed graphically and numerically.


The authors would like to thank the reviewers for their helpful suggestions and comments.

Conflict of Interest

The authors declare that there is no conflict of interests regarding the publication of this manuscript.


Muhammad Shoaib Sardar: https://www.orcid.org/0000-0001-7146-5639

Muhammad Asad Ali: https://www.orcid.org/0000-0002-9555-7885

Murat Cancan: https://www.orcid.org/0000-0002-8606-2274


How to cite this article: Muhammad Shoaib Sardar*, Muhammad Asad Ali, Faraha Ashraf, Murat Cancan. Study of middle graph for certain classes of graph by applying degree-based topological indices. Eurasian Chemical Communications, 2023, 5(1), 50-62. Link:  http://www.echemcom.com/article_155289.html


Copyright © 2023 by SPC (Sami Publishing Company) + is an open access article distributed under the Creative Commons Attribution License(CC BY)  license  (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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