Abstract
The area of graph theory (GT) is rapidly expanding and playing a significant role in cheminformatics, mostly in mathematics and chemistry to develop different physicochemical, chemical structure, and their properties. The manipulation and study of chemical graphical details are made feasible by using numerical structure invariant. Investigating these chemical characteristics of topological indices (TIs) is made possible by the discipline of mathematical chemistry. In this article, we study with the Cartesian product of complete graphs, with path graphs, and find their general result of connection number (CN)-based TIs, namely, first connection- based Zagreb index (1st CBZI), second connection- based Zagreb index (2nd CBZI), and third CBZI (3rd CBZI) and then modified first connection- based Zagreb index (CBZI) and second and third modified CBZIs. We also express the general results of first multiplicative CBZI, second multiplicative CBZI, and third and fourth multiplicative CBZI, of two special types of graphs, namely, complete graphs and path graphs. More precisely, we arrange the graphical and numerical analysis of our calculated expressions for both of Cartesian product with each other.
1. Introduction
The structural, biological, toxicological, and physicochemical features of existing chemical compounds are predicted by topological indices (TIs), which are numerical values associated with various chemical structures of molecular graphs. A molecular graph is a graph where the vertices stand in for atoms and the edges represent the covalent bonds connecting those atoms. TIs are widely utilised in the investigation of the relationships between the properties and activities of quantitative structures [1]. As one can see [1], several researchers have studied TIs. TIs are broken down into three categories: distance-based, degree-based, and polynomial-based. A TI that is based on the separation between the vertices is referred to as a distance-based TI. The basic TI was the Wiener index, which was created by Wiener in 1947, while he investigated the paraffin’s boiling point [2]. The creative idea of degree-based was created by Gutman [3]. After that, scientists looked into a range of distance-based descriptors in the chemical sector, which enabled them to understand chemical molecular data on chemical structures like freezing and melting point, density, stability, and flammability, among other properties (see [4, 5]). The degree of a vertex is what a degree-based TI is concerned with. Degree-based TIs are further divided into the two subgroups of connection-based TIs and degree-based TIs.
In 1975, Gutman and Trinajstic [6] investigated the novel idea of the first Zagreb index ( ZI). In addition, Gutman et al. [7] suggested the second Zagreb index ( ZI) in 1975. These traditional ZIs are crucial to the study of chemical network theory because of the wide range of applications they have. Later, Furtula and Gutman [8] put out the idea of third ZIs, also known as the forgotten index because it was not found until later. Modified ZI was examined by Nikolic et al. [9] in 2003. A new concept of connection number was recently introduced by Ali and Trinajstic [10], and it is the cardinality of those nodes that extend two lengths from a certain vertex. They stated that the connection-based Zagreb indices (CBZIs) had a greater capacity to predict the physical and chemical features of molecular structure than that of degree-based indices since they computed all the ZIs on connection bases as opposed to degree of the vertices. Recently, Liu et al. [11] provided Zagreb connection with a variety of molecular graphs based on operation in 2020. In addition, Javaid et al. [12] started new connection-based ZIs of various wheel-related graphs. Recently, Sattar et al. [13–15] computed CBZIs of different dendrimer nanostars.
Latinized scientists (cartesius) introduced the Cartesian coordinates in the 17th century. He provided the first systematic connection between algebra and geometry in a mathematical field. In 1912, Russell and Whitehead introduced the Cartesian product of graphs. They were frequently found later, notably by Gert Sabidussi in 1960. Imrich et al. [16] in 2008 studied the Cartesian product of some different types of graphs. Imrich and Peterin [17] proceeded with these evolutions to figure the TIs of Cartesian product of two graphs. Thereafter starting of CBZIs, many researchers started work on estimating the properties of Cartesian product of two graphs with the help of CBZIs. Klavzar and Rajapakse [18] just discovered the Cartesian product of different graphs. Furthermore, Imran and Shakila studied degree-based TIs for Cartesian product of F-sum connected graphs in 2017.
This composition is planned as follows: in Section 2, we discuss the fundamental concepts and basic definitions of graph theory. Section 3 contains the general expression to calculate the modified CBZIs of Cartesian product of graphs. Section 4 brings out the main results, and Section 5 includes conclusions. Table 1 shows the list all the acronyms used in this study.
2. Primary Definitions
In this section, we discuss the fundamental concepts and basic definition of graph theory.
Definition 1 (see [19]). We consider a simple and connected graph with vertex set and edge set , the order of , and the size of . The number of edges having as an last vertex is called degree of in graph and is denoted by .
Definition 2 (see [20]). We consider a graph ; the number of vertices at a distance of two from vertex is said to be connection number (CN) of vertex .
Definition 3 (see [21]). Let and be a graph with set of vertices and and set of edges and , respectively, then cartesian product is obtained by vertex set and .
Definition 4 (see [6]). let be a graph, where and represent the collection of edges and vertices, respectively. Following that, degree-based ZIs are defined as follows:(1)(2) where and show the degree of the vertex and , respectively
Definition 5 (see [10]). For a graph , CBZIs are given as follows:(1)(2) where and represent the of the vertex and , respectively. These CBZIs are known as 1st CBZI and CBZI, respectively.
Definition 6 (see [10, 22]). For a network , the modified CBZIs can be given as follows:(1)(2)(3)These modified CBZIs are known as the modified 1st CBZI, modified 2nd CBZI, and modified 3rd CBZI, respectively.
Definition 7 (see [12]). For a graph , 1st multiplactive CBZI, 2nd multiplactive CBZI, and 3rd and 4th multiplactive CBZI can be defined as follows:(1)(2)(3)(4)
Definition 8 (see [12]). For a graph , modified st multiplactive CBZI and modified 2nd and 3rd multiplactive CBZI can be defined as follows:(1)(2)(3)
3. Cartesian Product of Complete Graphs with Path Graphs
Let be a complete graphs of order with vertex set = and edge set.
= : and but . Now, let be a path graphs of order with vertex set = and edge set = .
The Cartesian product of . The edge set of is defined as follows: is adjacent if
In this section, we deal with Cartesian product of complete graph with path graph , where and s ≥ 6) are shown in Figure 1. In Figure 2, we discuss the structure of Cartesian product of complete graph with a path graph . For the first level, the first layer of is linked at every edges of , so the connection number is 4. So, for the next level, the second layer of is linked at every edges of , so then the connection number is 5. Furthermore, the process is repeated to obtain the next level and so on. We have also labelled the edges with their degree, and connection numbers are shown in Figure 2. In Tables 2 and 3, the edge partition of degree and connection number is presented separately. Now, we define the vertex partition of degree and connection number in this study, as shown in Tables 4 and 5.
Now, we do the partitioning of edge on the basis of connection number and degree of the Cartesian product of complete graphs with path graphs.
4. Main Results
Theorem 1. Let be a graph obtained by a Cartesian product of the complete graph and path graph , where and .
Proof. Using definition of 1st CBZI,
Theorem 2. Let be a graph obtained by a Cartesian product of the complete graph and path graph , where and .
Proof. Using definition of 2nd CBZI,
Theorem 3. Let be a graph obtained by a Cartesian product of the complete graph and path graph , where and .
Proof. Using definition of 1st modified CBZI,
Theorem 4. Let be a graph obtained by a Cartesian product of the complete graphs and path graph , where and .
Proof. Using definition of 2nd modified CBZI,
Theorem 5. Let be a graph obtained by the Cartesian product of the complete graph and path graph , where and .
Proof. Using the definition of 3rd modified CBZI,
Theorem 6. Let be a graph obtained by a Cartesian product of the complete graph and path graph , where and .
Proof. Using the definition of 1st multiplicative CBZI,
Theorem 7. Let be a graph obtained by the Cartesian product of the complete graph and path graph , where and .
Proof. Using the definition of 2nd multiplicative CBZI,
Theorem 8. Let be a graph obtained by the Cartesian product of the complete graph and path graph , where and .
Proof. Using the definition of 3rd multiplicative CBZI,
Theorem 9. Let be a graph obtained by the cartesian product of the complete graph and path graph , where and .
Proof. Using the definition of 4th multiplicative CBZI,
Theorem 10. Let be a graph obtained by the Cartesian product of the complete graph and path graph , where and
Proof. Using the definition of modified 1st multiplicative CBZI,
Theorem 11. Let be a graph obtained by the Cartesian product of the complete graph and path graph , where and .
Proof. Using the definition of modified 2nd multiplicative CBZI,
Theorem 12. Let be a graph obtained by the Cartesian product of the complete graph and path graph , where and .
Proof. Using the definition of the modified 3rd multiplicative CBZI,
5. Conclusion
In this study, we have final given expressions to calculate the TIs of the Cartesian product of complete graphs with path graphs . Scientists manipulate and examine the chemical organizational information with the use of TIs. In this study, we have calculated several TIs, including the 1st CBZI and 2nd CBZI. We have also computed modified 1st CBZI, 2nd CBZI, and 3rd CBZI and found 1st multiplicative CBZI, 2nd multiplicative CBZI, and 3rd multiplicative CBZI. Furthermore, a comparative analysis among all the calculated TIs is presented, which reveals the conclusion that modified 1st multiplicative CBZI, modified 2nd multiplicative CBZI, and modified 3rd multiplicative CBZI are more efficient to predict the physicochemical properties of chemical network. The researchers understand that the chosen structure will be made easier by this mathematical investigation, which will also motivate others to focus on the organic network.
Data Availability
The data used to support the findings of this study are included within this article. However, the reader may contact the corresponding author for more details on the data.
Additional Points
In future, we are interested to compute the Zagreb connection indices of the other type of graphs such as prism graphs and line graphs.
Conflicts of Interest
The authors declare that they have no conflicts of interest.