Abstract
The study of topological indices in graph theory is one of the more important topics, as the scientific development that occurred in the previous century had an important impact by linking it to many chemical and physical properties such as boiling point and melting point. So, our interest in this paper is to study many of the topological indices “generalized indices’ network” for some graphs that have somewhat strange structure, so it is called the cog-graphs of special graphs “molecular network”, by finding their polynomials based on vertex edge degree then deriving them with respect to , , and , respectively, after substitution of these special graphs are cog-path, cog-cycle, cog-star, cog-wheel, cog-fan, and cog-hand fan graphs; the importance of some types of these graphs is the fact that some vertices have degree four, which corresponds to the stability of some chemical compounds. These topological indices are first and second Zagreb, reduced first and second Zagreb, hyper Zagreb, forgotten, Albertson, and sigma indices.
1. Introduction
A graph is a pair order of vertex set and edge set where the cardinality of and are and , respectively. The degree of a vertex v represents the number of edges incident to that vertex and is denoted by dv. The maximum and minimum degrees of graph are denoted by and , respectively. The neighborhood of vertex , which is a set of all neighbors of and denoted by , is called open neighborhood, while the closed neighborhood of vertex , denoted by , is the set union the set . The degrees sum of neighbors of in is denoted by . For more information on many concepts in graph theory, see [1–3]. Chellali et al. were the first to introduce the vertex edge degree of graph [4]. The vertex edge degree (or degree) of vertex is equal to the number of elements in a set , and is denoted by the degree of vertex in . The maximum and minimum degrees of graph are denoted by and , respectively. Deutsch and Klavžar [5] first introduced the polynomial as follows:where is the number of edges such that . The polynomial is generally polynomial and may generate degree-based topological indices [6–10]. For applications in chemistry and networks, see [11–14]. Developed by Mondal et al., the polynomial is called the neighborhood polynomial and is defined as follows:where is the total number of edges such that . There are many recent works about neighborhood polynomials [15, 16]. Because of the importance of these topics, in 2023, Kavi et al. [17] proposed a new polynomial based on degree which is called polynomial, and it is defined as follows:where is the total number of edges such that . There are many recent works on degree and degree indices (see [18–20]).
In Table 1, we list many topological indices with respect to the vertex edge-degree of graph .
In this article, we calculate a closed form of some ve-degree dependent topological indices mentioned in Table 1 by applying some mathematical operation on Mve-polynomial of the graph, where the operators used are defined as follows:
, , , and . To illustrate the abovementioned concepts, we will take the following example, the polynomial, polynomial, and polynomial for a graph are shown in Figure 1, respectively:(i)(ii)(iii)
In Table 2, we give some vertex-edge-degree-based topological indices for the graph in Figure 1.
2. Polynomial of Some Cog-Special Graphs
In this section, we find the polynomials and indices for some cog-special graphs, such as path, cycle, star, complete, and wheel graphs [21].
2.1. Cog-Path Graph
Definition 1. A cog-path graph is the graph constructed from a path , , of a set vertex , and with additional vertices , and edges , as shown in Figure 2.
2.1.1. Some Properties of a Cog-Path Graph
(i)The order and the size are and , respectively(ii)The degrees of vertices , are which represent the maximum degree “” and the degrees of vertices , , and ; are which represent the minimum degree “”(iii)The maximum and minimum degrees are and , respectivelyIn the following theorem, we will find the polynomial of the cog-path graph, , .
Theorem 2. Let be the cog-path graph of order , . Then,
Proof. From Definition 1, the vertex set is , where and . Since the vertices , , , and are adjacent to , for , the two vertices and are adjacent and the two vertices and are adjacent, thenHence, for all such that , we have
Remark 3. (i)(ii).In the next corollary, we can easily get the topological indices of , , and from the derivatives with respect to , , and , respectively, after substitution .
Corollary 4. Let be the cog-path graph of order , , then we have(1)(2)(3)(4)(5)(6)(7)(8).
2.2. Cog-Cycle Graph
Definition 5. Let ,, be a cycle of order , The cog-cycle is obtained from by adding new vertices and edges , , as shown in Figure 3.
2.2.1. Some Properties of a Cog-Cycle Graph
(i)The order and the size are and , respectively(ii)The degrees of vertices , are which represent the maximum degree “” and the degrees of vertices , are which represent the minimum degree “”(iii)The maximum and minimum degrees are and , respectivelyTheorem 6. Let be the cog-cycle graph of order , . Then,
Proof. From Definition 5, the vertex set is , where and . Since the vertices , , , and are adjacent to , for , where , , and , thenHence, for all such that , we have
Remark 7. From easy to obtain many indices topologically in the following corollary by the derivatives with respect to , , and , respectively, after substitution for .
Corollary 8. Let be the cog-cycle graph of order , , then we have(1)(2)(3)(4)(5)(6)(7)(8)
2.3. Cog-Star Graph
Definition 9. A cog-star graph is the graph constructed from a star graph , , of a vertex set with additional vertices , and edges , , as shown in Figure 4.
2.3.1. Some Properties of a Cog-Star Graph
(i)The order and the size: and , respectively(ii)The degree of vertex is which represent the maximum degree “,” the degrees of vertices , are and the degrees of vertices , are which represent the minimum degree “”(iii)The maximum and minimum degrees are and , respectivelyTheorem 10. Let be the cog-star graph of order , . Then,
Proof. Since the vertex is adjacent to , for and every vertices are adjacent to and , for , where , see Figure 4. Then,Hence,
Corollary 11. Let be a cog-star graph of order , , then we have(1)(2)(3)(4)(5)(6)(7)(8)
2.4. Cog-Complete Graph
Definition 12. A cog-complete graph is the graph constructed from a complete graph , of a vertex set with additional vertices , and edges , , as shown in Figure 5.
2.4.1. Some Properties of a Cog-Complete Graph
(i)The order and the size are and , respectively(ii)The degrees of vertices , are which represent the maximum degree “” and the degrees of vertices , are which represent the minimum degree “”(iii)The maximum and minimum degrees are and , respectivelyTheorem 13. Let be the cog-complete graph of order , . Then,
Proof. For all , the vertex is adjacent to , the vertex is adjacent to , and the vertex is adjacent to , thenHence, for all such that , we have
Remark 14. It is clear that .
Corollary 15. Let be a cog-complete graph of order , , then we have(1)(2)(3)(4)(5)(6)(7)(8)
2.5. Cog-Wheel Graph
Definition 16. A cog-wheel graph is the graph constructed from a wheel , , of a vertex set , and with additional vertices , and edges , , as shown in Figure 6.
2.5.1. Some Properties of a Cog-Wheel Graph
(i)The order and the size are and , repectively(ii)The degree of vertex is which represent the maximum degree “,” the degrees of vertices , are and the degrees of vertices , are which represent the minimum degree “”(iii)The maximum and minimum degrees are and , respectivelyTheorem 17. Let be the cog-wheel graph of order , . Then,
Proof. Since the vertex is adjacent to , for , the vertex is adjacent to two adjacent vertices and , for , where , see Figure 6, thenHence, for all such that , we have
Remark 18. (i)(ii).In the next corollary, we can easily obtain the topological indices of the from the derivative with respect to , , and by the substitution .
Corollary 19. Let be a cog-wheel graph of order , , then we have(1)(2)(3)(4)(5)(6)(7)(8)
3. Polynomial of Two Types of Cog-Fan Graphs
In this section, we find polynomials of two types of cog-fan graphs, such as fan and hand fan graphs [17].
3.1. Cog-Fan Graph
Definition 20. A cog-fan graph is a graph constructed from a fan graph , ( is an even number) of a vertex set by adding vertices and edges to the fan graph , see Figure 7.
3.1.1. Some Properties of a Cog-Fan Graph
(i)The order and the size are and , respectively(ii)The degree of vertex is which represent the maximum degree “,” the degrees of vertices , are and the degrees of vertices , are which represent the minimum degree “”(iii)The maximum and minimum degrees are and , respectivelyTheorem 21. Let be the cog-fan graph of order , and is even. Then,
Proof. Since the vertices and are adjacent to and , for all , thenHence,
Corollary 22. Let be the cog-fan graph of order , . Then we have(1)(2)(3)(4)(5)(6)(7)(8)
3.2. Cog-Hand Fan Graph
Definition 23. A cog-hand fan graph is the graph constructed from a hand fan graph , of vertex set by adding vertices and edges to the graph , see Figure 8.
3.2.1. Some Properties of a Cog-Hand Fan Graph
(i)The order and the size are and , respectively(ii)The degree of vertex is which represent the maximum degree “,” the degree of vertices and are , the degrees of vertices , are and the degrees of vertices , are which represent the minimum degree “”(iii)The maximum and minimum degrees are and , respectivelyTheorem 24. Let be the cog-hand fan graph of order , . Then,
Proof. Since the vertex is adjacent to , for all . Also, the vertex is adjacent to two adjacent vertices and , for all , then,Hence, for all such that , we have
Remark 25.
Corollary 26. Let be a cog-hand fan graph of order 2 , . Then we have(1)(2)(3)(4)(5)(6)(7)(8)
4. Conclusions
The topological indices obtained by presenting a new definition of the polynomial that depends on the degree of a vertex , as presented by Chellali et al., are distinguished by the fact that they lie between the topological indices of degrees of a vertex and the topological indices of the neighborhoods of vertex . We noticed that the topological indicators mentioned in Table 1 for both the cog-path and the cog-cycle have the same slope, only the start points are different. In addition to that, the first Zagreb and reduced first Zagreb indices are almost identical, and the Albertson and sigma indices are also almost identical in all cog-path and cog-cycle graphs. The first Zagreb and reduced first Zagreb indices are almost identical, the second Zagreb and reduced second Zagreb indices are almost identical, and the Albertson index is very close to both the second Zagreb and reduced second Zagreb indices in all cog-star, cog-wheel, cog-complete, cog-fan, and cog-hand fan graphs with very few differences, see Figure 9. Finally, the topological indices mentioned in Table 1 have the same behavior in all the private graphs studied in the paper.
Data Availability
The data used to support the findings of this study are available at the corresponding author upon request.
Conflicts of Interest
The authors declare that they do not have any conflicts of interest.
Acknowledgments
This paper was supported by the Faculty of Science, University of Zakho, College of Basic Education, University of Salahaddin and College of Computer Science and Mathematics, University of Mosul.