The <i>Y</i>-Index of Some Complement Graph Structures and Their Applications of Nanotubes and Nanotorus

Alsharafi, Mohammed; Alameri, Abdu; Zeren, Yusuf; Shubatah, Mahioub; Alwardi, Anwar

doi:https://doi.org/10.1155/2024/4269325

Journal of Mathematics

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Research Article | Open Access

Volume 2024 | Article ID 4269325 | https://doi.org/10.1155/2024/4269325

The Y-Index of Some Complement Graph Structures and Their Applications of Nanotubes and Nanotorus

Mohammed Alsharafi,^1,2Abdu Alameri ,³Yusuf Zeren,¹Mahioub Shubatah,²and Anwar Alwardi⁴

Academic Editor: G. Muhiuddin

Received07 Aug 2022

Revised15 Oct 2022

Accepted05 May 2023

Published17 Jan 2024

Abstract

Topological descriptors play a significant role in chemical nanostructures. These topological measures have explicit chemical uses in chemistry, medicine, biology, and computer sciences. This study calculates the Y-index of some graphs and complements graph operations such as join, tensor and Cartesian and strong products, composition, disjunction, and symmetric difference between two simple graphs. Moreover, the Y-polynomial of titania nanotubes and the formulae for the Y-index, Y-polynomial, F-index, F-polynomial, and Y-coindex of the and nanotubes and their molecular complement graphs have been investigated.

1. Introduction

Topological indices are useful molecular descriptions in the field of chemical graph theory to establish the structural property and structural-activity relationship that describes chemical component structures and helps predict certain chemical-physical properties [1, 2]. There are many topological descriptors that are proposed and studied based on degree, distance, eigenvalue, matching, and mixed and other parameters of graphs [3].

The importance of computing the complement graphs is to identify adjacent intervals and thus develop the data structure to process them efficiently [4]. There have been many studies that have improved the complexity of time problems related to dense graphs and complement graphs (see [5–7]). On some problems, the computational time may be reduced by using the algorithms of complement graphs. For example, Ito and Yokoyama explored the linear time resolution of several storage-based problems [8]. They have looked at storage methods for representing nondirected graphs and maintaining the graphs and their complement graphs in the data structure. They demonstrated that the order of legal nodes and sparse subgraphs that preserve the connectivity properties of a specific graph of the complement graph can be found in linear time and that the width-first search tree and the depth-first search tree of the complement graph of a specific graph can be constructed in linear time.

In our study, we consider a finite connected and an undirected graph with edges and vertices. The vertex degree of is the number of edges connected to and is represented by . The size of a graph is the number of edges in and is expressed as and the number of vertices of is called the order of and is represented by . The complement of a graph , indicated by , is a graph on the same set such that every two vertices and are adjacent to each other, i.e., they are connected by an edge if, and only if, they are not adjacent to each other in . Then, . Consequently, and , the degree of a vertex in , is the number of edges connected to and is defined by . For example, the benzene graph and its complement graph are shown in Figure 1. The 1^st and 2^nd Zagreb indices are considered to be one of the oldest descriptors of the graph defined in 1972 by Gutman and Trinajstić [9]. They are defined for a graph as follows:

Došlić [10] defined Zagreb coindices as follows:

Furtula et al. in 2015 [11, 12] presented the Forgotten index (F-index) that is defined as follows:

De et al. [13] introduced a new descriptor denoted by F-coindex, defined as follows:

Alameri et al. [14, 15] in 2020 defined new degree-based descriptors, denoted by the and , and they are, respectively, defined as follows:

Also, in the same papers, the and formulae of the graph and complement graph are investigated and defined as follows:

The first general Zagreb index introduced by Li and Zheng is as follows [16]:

Several studies have been conducted on various topological indices in different graph operations. De et al. [13] introduced the F-index and the F-coindex in several graph operations, such as join, union, Cartesian and corona products, composition, tensor and strong products, symmetric difference, and disjunction in graphs. Another study by Veylaki et al. [17] in 2015 derived some exact formulae for calculating the third and hyper-Zagreb coindices of certain graph operations. Khalifeh et al. [18] and Ashrafi et al. [19], respectively, computed the 1^st and 2^nd Zagreb indices and coindices of some operations on graphs. Das et al. [20] obtained some exact formulae for computing upper bounds for multiplicative Zagreb indices for some operations on graphs. In 2014, De et al. [21], in different graph operations, have obtained explicit formulae of the connective eccentric index. Azari and Iranmanesh [22] presented exact formulae for the eccentric distance sum of some operations on graph. Alameri et al. [14, 15] computed the Y-index and coindex of the Cartesian product , composition , disjunction , symmetric difference , tensor product , and strong product of two undirected and connected graphs. Alsharafi et al. [23–25] studied the first, second, forgotten, and second hyper-Zagreb indices of some graph and complement operations on the graph. This paper will further explore the behavior of the Y-index for joining two connected graphs and the Y-index of the various graph operations to supplement the graph and apply the results to find the Y-index of some certain nanostructures. However, many other graphs operations do not cover here. For further research, it is possible to consider the Y-index and Y-coindex of various other graphs and the complement graph operations.

Now, we present a few definitions of operations that we would be using in our results as follows:

The tensor product of and graphs is the graph with the vertex, set where is adjacent with iff is adjacent with and is adjacent with .

The join of and graphs with disjoint and sets and edge and sets is the union graph, both with all the edges that join and with each other.

The Cartesian product of and graphs with disjoint and vertex sets and and edge sets is the graph with the vertex set , where is adjacent with whenever and is adjacent with or and is adjacent with such that and .

The composition of and graphs with disjoint and vertex sets and and edge sets is the graph with the vertex set, where is adjacent with , whenever is adjacent with or and is adjacent with such that and .

The strong product of and graphs with and vertex sets and and edge sets is the graph with the vertex set , where is adjacent with whenever and is adjacent with or and is adjacent with or is adjacent with and is adjacent with .

The disjunction of and graphs with and vertex sets and and edge sets is the graph with the vertex set, where is adjacent with whenever or .

The symmetric difference of and graphs with vertex and sets and and edge sets is the graph with the vertex set , where is adjacent with whenever or but not both.

2. Main Results

In this section, we examine the Y-index for the join of two graphs and the Y-index of the binary operations of various complement graphs such as the tensor and the Cartesian and strong product, the join, the composition, the disjunction, and the symmetric difference of the two graphs. In addition, we obtained the formulae for the Y-polynomial and Y-index of some nanotubes and nanotorus and their molecular complement graph.

2.1. The Y-Index of Some Complement Graph Operations

In this subsection, we compute the Y-Index of various complement graph operations as the following theorems.

Theorem 1. Let and be two two undirected connected graphs with vertices and edges, then

Proof. By equation (6) and since , , and are given in [14, 26, 27], respectively, then

Theorem 2. The of is given by

Proof. By definitions of the join and the , we have

Theorem 3. Let and be two undirected connected graphs with vertices and edges, then

Proof. From Theorem 2 and since and are given in [26, 27], respectively, and applying equation (6), we get the required.

Theorem 4. The of is given by

Proof. Since , , and are given in [18, 28, 29], respectively, and applying equation (6), we get the required.

Theorem 5. Let and be two undirected connected graphs with vertices and edges, then

Proof. Since , and are given in [14, 18, 27], respectively, and applying equation (6), we get the required.

Theorem 6. The of is given by

Proof. By [27, 28, 30], respectively, we haveAlso, by applying equation (6), we get the required.

Theorem 7. The of is given by

Proof. By [14, 27, 31], respectively, we haveAlso, by applying equation (6) for , we get the required.

Theorem 8. The of complement is given by

Proof. By using the same method such as in Theorem 7, we get the required.

2.2. Y-Polynomial of Titania Nanotubes and Y-Index of Molecular Complement Titania Nanotubes

Titania nanotubes are semiconductors that have been systematically synthesized and studied carefully as potential technological materials [32–34], and some authors have calculated some of its topological indices [32–35]. In the following theorems and corollaries, we obtained the formula for of titania nanotubes and computed the general expressions for the of complement structure of titania nanotubes.

Theorem 9. Let be titania nanotubes (see Figure 2). Then, of is given by

(a)

(b)

Proof. Since the definition of Y-polynomial of the graph is , the vertex and edge partitions of titania nanotube are given as Corollary 4.1 in [30], and the edge set of Ψ is divided into four edges, depending on the degree of the vertices as follows:Therefore,

Corollary 10. The Y-index of complement nanotube is given by

Proof. From [32, 36], we have , , and , and the collection of the cardinality of the vertex and edge sets of titania nanotubes are given as .
Also, by applying equation (6) for , we have

2.3. Y-Index of Nanotube

In this subsection, we present some formulae for the and of the nanotubes and its molecular complement graph. Moreover, we apply on the line graphs of the nanotubes.

Theorem 11. Let nanotube and the of (Figure 3) is given by

Proof. The vertex and edge partitions and of are given in Tables 1 and 2, respectively.
The edge set of for the sum degree on the neighbors of each vertex can be divided into six separate edges as follows:Therefore, by using the definition of the , we have

Corollary 12. The of nanotube (Figure 3) is given by

Proof. By the definition of the and Theorem 11, we have

Theorem 13. The forgotten index of nanotube (Figure 3) is given by

Proof. Using the definition of the , we get

Theorem 14 (see [37]). The 1^st 2^nd Zagreb indices and hyper-Zagreb index of nanotube are given by

Corollary 15. The of nanotube (Figure 3) is given by

Proof. By the definition of the and Theorem 13, we have

Corollary 16. The of the complement nanotube (Figure 3) is given by

Proof. From Theorems 11 and 13, , , and ; moreover, the collection of the cardinality of the vertex and edge sets of nanotube areAlso, by applying equation (6) for , we obtain

Theorem 17. The of nanotube (Figure 3) is given by

Proof. Using Theorems 11 and 13 and applying equation (7), we get the required.

2.4. Y-Index of Nanotube

In this subsection, we compute the and of the nanotubes and its molecular complement graph. Moreover, we apply on the line graphs of the nanotubes.

Theorem 18. Let , then the of (Figure 3) is given by

Proof. The vertex and edge partitions and of are given in Tables 3 and 4, respectively.
The edge set of can be divided into six disjoint edge sets as follows:Therefore, by using the definition of the , we have

Corollary 19. The of nanotube (Figure 3) is given by

Proof. By the definition of the and Theorem 18, we have

Theorem 20. Let nanotube, then the forgotten index of (Figure 3) is given by

Proof. By using the definition of the , we get

Theorem 21 (see [37]). The 1^st and 2^nd Zagreb indices and hyper-Zagreb index of nanotube are given by

Corollary 22. The of nanotube (Figure 3) is given by

Proof. By the definition of the and Theorem 20, we obtain

Corollary 23. The of the complement nanotube (Figure 3) is given by

Proof. From Theorems 18 and 20, there are , , and ; furthermore, the collections of the cardinality of the vertex and edge sets of nanotube areAlso, by applying equation (6) for , we have

Theorem 24. The of nanotube (Figure 3) is given by

Proof. Using Theorems 18 and 20 and applying equation (7), we get the required.

Corollary 25. Assume that is the molecular structure of a nanotorus (see Figure 4), then the of complement q-multiwalled nanotorus is given by(a)(b)

Proof. Proving item (a) by applying equation (6) for and since , , , are given in [29, 38, 39], respectively, thenProving item (b) by definition of Cartesian product, we have and , , and are given in [24, 38, 39], respectively, and applying Theorem 4, we get the required.

2.5. Numerical and Graphical Representation

In this subsection, using MATLAB (R2022b), the numerical values of the 1^st and 2^nd Zagreb indices, F-index, hyper-Zagreb index, and Y-index of and nanotubes have been computed. The numerical representation is depicted in Tables 5 and 6. In Table 5, data analysis of some indices’ values of nanotubes are presented and formulae are reported in Theorems 11, 13, and 14 for the nanotubes. In Table 6, some indices’ values of nanotubes are presented and formulae are reported in Theorems 18, 20, and 21 for nanotubes. In both tables, it shows that values of 1^st and 2^nd Zagreb indices, forgotten, hyper-Zagreb, and Yemen indices are in increasing order as the values of q, p increase.

The graphical representations are shown in Figures 5–9. In Figures 7–9, numerical comparison of Y-polynomial of titania nanotubes and and nanotubes are presented. In Figures 5 and 6, a comparison of values of some widely known topological indices is presented. We can easily see that (), the 1^st and 2^nd Zagreb indices, F-index, hyper-Zagreb index, and Y-index of and nanotubes are in increasing order as the values of are increasing. However, the 1^st and 2^nd Zagreb indices have the least prediction potentials with respect to the dimension, while the hyper-Zagreb index and Y-index increase quickly as a function of dimensions.

3. Conclusions

In light of our analysis of structures and mathematical derivations, this paper has uncovered Y-index formulae for various graph and complement graph operations. These operations include tensor and Cartesian and strong products, join, composition, disjunction, and symmetric difference of two graphs. Moreover, we have computed the Y-polynomial of titania nanotubes and the Y-index of the molecular complement graph of titania nanotubes and nanotorus. Furthermore, we have investigated the Y-index, Y-polynomial, F-index, F-polynomial, and Y-coindex formulae of and nanotubes and their molecular complement graphs. To provide further insight, we have presented numerical comparisons of our computed results and illustrated corresponding graphical behavior through figures. While this research has provided valuable contributions to the field, much work remains to be done. As such, we have outlined several potential directions for future research, including the eigenvalue-based, matching-based, and mixed-based indices of nanotubes and nanotorus.

Data Availability

The data used to support the findings of this study are included within the manuscript.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

S. Mondal, A. Bhosale, N. De, and A. Pal, “Topological properties of some nanostructures,” Nanosystems: Physics, Chemistry, Mathematics, vol. 11, no. 1, pp. 14–24, 2020.
View at: Publisher Site | Google Scholar
B. Furtula, I. Gutman, and M. Dehmer, “On structure-sensitivity of degree-based topological indices,” Applied Mathematics and Computation, vol. 219, no. 17, pp. 8973–8978, 2013.
View at: Publisher Site | Google Scholar
S. D. Stankov, M. M. Matejić, I. Ž. Milovanović, E. I. Milovanović, and S. B. Altındağ, “Some new bounds on the first Zagreb index,” Electronic Journal of Mathematics, vol. 1, pp. 101–107, 2020.
View at: Google Scholar
M. Y. Kao, N. Occhiogrosso, and S. H. Teng, “Simple and efficient graph compression schemes for dense and complement graphs,” Journal of Combinatorial Optimization, vol. 2, no. 4, pp. 351–359, 1998.
View at: Publisher Site | Google Scholar
D. Gusfield, “A graph theoretic approach to statistical data security,” SIAM Journal on Computing, vol. 17, no. 3, pp. 552–571, 1988.
View at: Publisher Site | Google Scholar
M. Y. Kao, “Data security equals graph connectivity,” SIAM Journal on Discrete Mathematics, vol. 9, no. 1, pp. 87–100, 1996.
View at: Publisher Site | Google Scholar
M. Y. Kao and D. Gusfield, “Efficient detection and protection of information in cross tabulated tables I: linear invariant test,” SIAM Journal on Discrete Mathematics, vol. 6, no. 3, pp. 460–476, 1993.
View at: Publisher Site | Google Scholar
H. Ito and M. Yokoyama, “Linear time algorithms for graph search and connectivity determination on complement graphs,” Information Processing Letters, vol. 66, no. 4, pp. 209–213, 1998.
View at: Publisher Site | Google Scholar
I. Gutman and N. Trinajstić, “Graph theory and molecular orbitals. Total φ-electron energy of alternant hydrocarbonsπ-electron energy of alternant hydrocarbons,” Chemical Physics Letters, vol. 17, no. 4, pp. 535–538, 1972.
View at: Publisher Site | Google Scholar
T. Došlić, “Vertex-weighted Wiener polynomials for composite graphs,” Ars Mathematica Contemporanea, vol. 1, pp. 66–80, 2008.
View at: Publisher Site | Google Scholar
B. Furtula and I. Gutman, “A forgotten topological index,” Journal of Mathematical Chemistry, vol. 53, no. 4, pp. 1184–1190, 2015.
View at: Publisher Site | Google Scholar
B. Futrula, I. Gutman, Ž. K. Vukicevic, G. Lekishvili, and G. Popivoda, “On an old/new degree-based topological index,” Bulletin (Academie Serbe Des Sciences Et Des Arts Classe Des Sciences Mathematiques Et Naturelles Sciences mathematiques), vol. 40, pp. 19–31, 2015.
View at: Google Scholar
N. De, S. Nayeem, and A. Pal, “The F-coindex of some graph operations,” SpringerPlus, vol. 5, no. 1, 2016.
View at: Publisher Site | Google Scholar
A. Alameri, N. Al-Naggar, M. Al-Rumaima, and M. Alsharafi, “Y-index of some graph operations,” International Journal of Applied Engineering Research, vol. 15, no. 2, pp. 173–179, 2020.
View at: Google Scholar
A. Alameri, M. Al-Rumaima, and M. Almazah, “Y-coindex of graph operations and its applications of molecular descriptors,” Journal of Molecular Structure, vol. 1221, Article ID 128754, 2020.
View at: Publisher Site | Google Scholar
X. Li and J. Zheng, “A unified approach to the extremal trees for different indices,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 54, no. 1, pp. 195–208, 2005.
View at: Google Scholar
M. Veylaki, M. Nikmehr, and H. Tavallaee, “The third and hyper-Zagreb coindices of some graph operations,” Journal of Applied Mathematics and Computing, vol. 50, no. 1-2, pp. 315–325, 2015.
View at: Publisher Site | Google Scholar
M. Khalifeh, H. Yousefi-Azari, and A. R. Ashrafi, “The first and second Zagreb indices of some graph operations,” Discrete Applied Mathematics, vol. 157, no. 4, pp. 804–811, 2009.
View at: Publisher Site | Google Scholar
A. Ashrafi, T. Došlić, and A. Hamzeh, “The Zagreb coindices of graph operations,” Discrete Applied Mathematics, vol. 158, no. 15, pp. 1571–1578, 2010.
View at: Publisher Site | Google Scholar
K. C. Das, A. Yurttas, M. Togan, A. S. Cevik, and I. N. Cangul, “The multiplicative Zagreb indices of graph operations,” Journal of Inequalities and Applications, vol. 2013, pp. 14–90, 2013.
View at: Publisher Site | Google Scholar
N. De, A. Pal, and S. Nayeem, “On some bounds and exact formulae for connective eccentric indices of graphs under some graph operations,” International Journal of Combinatorics, vol. 2014, Article ID 579257, 5 pages, 2014.
View at: Publisher Site | Google Scholar
M. Azari and A. Iranmanesh, “Computing the eccentric-distance sum for graph operations,” Discrete Applied Mathematics, vol. 161, no. 18, pp. 2827–2840, 2013.
View at: Publisher Site | Google Scholar
M. Alsharafi, M. Shubatah, and A. Alameri, “The first and second Zagreb index of complement graph and its applications of molecular graph,” Asian Journal of Probability and Statistics, vol. 8, no. 3, pp. 15–30, 2020.
View at: Publisher Site | Google Scholar
M. Alsharafi, M. Shubatah, and A. Alameri, “The forgotten index of complement graph operations and its applications of molecular graph,” Open Journal of Discrete Applied Mathematics, vol. 3, no. 3, pp. 53–61, 2020.
View at: Publisher Site | Google Scholar
M. Alsharafi, Y. Zeren, and A. Alameri, “The second hyper-zagreb index of complement graphs and its applications of some Nano structures,” Asian Journal of Probability and Statistics, vol. 15, no. 4, pp. 54–75, 2021.
View at: Publisher Site | Google Scholar
K. Kiruthika, “Zagreb indices and Zagreb coindices of some graph operations,” International Journal of Advanced Research in Engineering & Technology, vol. 7, no. 3, pp. 25–41, 2016.
View at: Google Scholar
N. De, S. M. A. Nayeem, and A. Pal, “F-Index of some graph operations,” Discrete Mathematics, Algorithms and Applications, vol. 8, no. 2, Article ID 1650025, 2016.
View at: Publisher Site | Google Scholar
M. Alsharafi and M. Shubatah, “On the hyper-zagreb index of some graph binary operations,” Asian Research Journal of Mathematics, vol. 16, no. 4, pp. 12–24, 2020.
View at: Publisher Site | Google Scholar
H. Aram and N. Dehgardi, “Reformulated F-index of graph operations,” Communications in Combinatorics and Optimization, vol. 1, no. 2, pp. 87–98, 2017.
View at: Google Scholar
M. Alsharafi, M. Shubatah, and A. Alameri, “The hyper-Zagreb index of some complement graphs,” Advances in Mathematics: Scientific Journal, vol. 9, no. 6, pp. 3631–3642, 2020.
View at: Publisher Site | Google Scholar
A. Alameri, M. Alsharafi, E. Ali, and M. Gumaan, “A note on Topological indices and coindices of disjunction and symmetric difference of graphs,” Discrete Applied Mathematics, vol. 304, pp. 230–235, 2021.
View at: Publisher Site | Google Scholar
M. A. Malik and M. Imran, “On multiple Zagreb indices of TiO2 nanotubes,” Acta Chimica Slovenica, vol. 62, no. 4, pp. 973–976, 2015.
View at: Publisher Site | Google Scholar
I. Nadeem and H. Shaker, “On eccentric connectivity index of TiO2 nanotubes,” Acta Chimica Slovenica, vol. 63, no. 2, pp. 363–368, 2016.
View at: Publisher Site | Google Scholar
N. De, “On molecular topological properties of TiO2 nanotubes,” Journal of Nanoscience, vol. 2016, pp. 1–5, 2016.
View at: Publisher Site | Google Scholar
A. Alameri, “Second hyper-zagreb index of Titania nanotubes and their applications,” IEEE Access, vol. 9, pp. 9567–9571, 2021.
View at: Publisher Site | Google Scholar
D. Nilanjan, “On molecular topological properties of TiO2 nanotubes,” Hindawi, Journal of Nanoscience, vol. 5, 2016.
View at: Google Scholar
Z. Shao, M. K. Siddiqui, and M. H. Muhammad, “Computing Zagreb indices and Zagreb polynomials for symmetrical nanotubes,” Symmetry, vol. 10, no. 7, p. 244, 2018.
View at: Publisher Site | Google Scholar
M. Alsharafi, M. Shubatah, and A. Alameri, “On the Hyper-Zagreb coindex of some Graphs,” The Journal of Mathematics and Computer Science, vol. 10, no. 5, pp. 1875–1890, 2020.
View at: Google Scholar
A. Ayache, A. Alameri, M. Alsharafi, and H. Ahmed, “The second hyper-zagreb coindex of chemical graphs and some applications,” Journal of Chemistry, vol. 2021, Article ID 3687533, 8 pages, 2021.
View at: Publisher Site | Google Scholar

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Copyright © 2024 Mohammed Alsharafi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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