Abstract
In this paper, we investigate square-hexagonal chains, a class of systems where the inner dual of a structure with a square-hexagon shape forms a path graph. The specific configuration of square and hexagonal polygons, and how they are concatenated, leads to different types of square-hexagonal chains. A square containing a vertex of degree 2 is classified as having a kink, and the resulting kink is referred to as a type kink. This kink is further subdivided into three types: , 2, and 2. We focus on the kink chain of type 2 and compute various topological descriptors for this configuration. By deriving analytical expressions, we determine the maximizing and minimizing values of these descriptors. Additionally, we provide a comprehensive analysis of the expected values for these descriptors and offer a comparison of their behaviors through analytical, numerical, and graphical methods. These results offer insights into the structural properties and behavior of square-hexagonal chains, particularly in relation to the optimization of topological descriptors.
1 Preliminaries
Graph theory is a mathematical discipline that studies graphs, which are abstract structures used to model and analyze relationships between objects. A Graph is defined to be the collection of vertices (nodes) and edges (links or arcs) where and denotes set of vertices and edges respectively. The order of a graph, denoted as refers to the total number of vertices in the graph. The size of a graph, denoted as refers to the total number of edges in the graph. The degree of a vertex, denoted by or , is the number of edges connected to that vertex. The distance between two vertices is the length of the shortest path joining them. For basic definitions related to graph theory, we refer (Trinajstic, 1992).
In chemical graph theory, the numerical values assigned to a molecular graph, known as topological indices or molecular descriptors, are often used to correlate with chemical structures and their properties. In other words, topological indices refer to graph invariants or descriptors that have significant chemical relevance. These indices are based on the graphical representation of a molecule and can encode chemical information such as atom types and bond multiplicities. Topological indices are valuable for predicting specific chemical and physical properties of the underlying molecular structure, combining logical and mathematical principles to translate a molecule’s symbolic representation into a useable numerical form. Chemical graph theory, which merges the fields of chemistry and graph theory, uses graphs to represent chemical structures, providing insights into the physical and chemical characteristics of molecules.
The first degree based topological descriptor was introduced by Milan Randic in 1975 in his paper (Randic, 1975) “On characterization of molecular branching.” This index is referred to as Randic index and is defined asThe Randic index has been recognized as a valuable tool in drug design and has been widely used for this purpose in various studies (Randic, 1975).
The first and second Zagreb indices are the oldest degree based graph invariants introduced by Gutman and Trinajstic (Gutman and Trinajstic, 1972b) in 1972. They were later included among topological descriptors and are defined as
The first and second Zagreb indices were initially applied to branching problem (Gutman et al., 1975b). Later, they found applications in QSPR and QSAR studies (Balaban, 1979; Bonchev and Trinajsti, 2001; Devillers and Balaban, 1999).
The applicability of Zagreb indices motivated the researchers to define different variants of Zagreb indices. The Hyper Zagreb index was put forwarded by Shirdel et al. (Shirdel et al., 2013) in 2013 and is defined as
Another variant of Zagreb indices namely, first and second redefined Zagreb indices were introduced by Ranjini et al. (Ranjini et al., 2013)
Motivated by the definitions of first and second Zagreb indices and their chemical applicability, V. Kulli (Kulli, 2017a) introduces the first and second Gourava indices. These topological indices are defined as
The first and second Revan descriptors were introduced by V. Kulli (Kulli, 2017b) and are defined aswhere is defined as , where and denotes the maximum and minimum degree among the vertices of .
For more details on the importance of topological indices and their applications see (Noreen and Mahmood, 2018; Wei and Shiu, 2019; Raza, 2020; Wei et al., 2020; Fang et al., 2021; Alraqad et al., 2022; Zhang X. et al., 2023; Zhang Guoping et al., 2023; Hui et al., 2023a; Hui et al., 2023b; Huang et al., 2023). For results related to mathematical properties of the topological indices, we refere (Zhou, 2004; Zhao et al., 2016; Gao et al., 2017; Kulli, 2017c; Kulli, 2017e; Zhang et al., 2024; Govardhan et al., 2024; Prabhu et al., 2024a; Prabhu et al., 2024b).
2 Square-hexagonal system and kink chains
A square-hexagonal system, also known as a rectangular hexagonal system, is a connected geometric structure created by joining equal-sized squares and hexagons. This arrangement blends elements of square and hexagonal lattices, forming a distinctive repeating pattern that combines the characteristics of both shapes. The lattice points in this system create a regular and continuous design, where each polygon is linked to its neighbors. Two polygons are considered neighboring if they share a common edge, emphasizing the interconnected nature of this hybrid structure. This system is widely used in crystallography and materials science, particularly for analyzing the structures of materials with hexagonal crystal systems that exhibit square symmetry along specific crystallographic directions. It provides a geometric framework for understanding the arrangement of atoms, ions, or other structural components within such materials.
A square-hexagonal system is a two-dimensional lattice structure that combines square and hexagonal elements in a unified arrangement. In contrast, a square-hexagonal chain is a one-dimensional sequence where square and hexagonal configurations alternate along its length. While both concepts incorporate square and hexagonal features, they differ in their structure and intended applications. The structure of a square-hexagonal chain varies depending on the types of polygons used and how they are concatenated. A square-hexagonal chain composed of polygons is denoted as . If all the polygons in are squares, it is referred to as a polyomino chain (Li et al., 2023). Similarly, if all the polygons are hexagons, is called a hexagonal chain (Alraqad et al., 2022). However, when squares and hexagons alternate in the chain, is specifically known as a phenylene chain. (as in (Raza, 2021; Shooshtari et al., 2022).
To derive key results, it is important to introduce certain terminologies related to square-hexagonal chains. In graph theory, a kink refers to a point in the graph where there is a sudden change in direction or slope. More precisely, a kink is a vertex whose degree is greater than the degrees of its neighboring vertices, resulting in a bend or angular deviation in the graph’s structure.
Kinks play a significant role in graph analysis, as they often highlight structural changes or key points within the graph. They can influence various graph properties and algorithms, including traversal methods, connectivity analysis, and the identification of critical nodes or hubs in networks. In network analysis, for instance, identifying kinks or high-degree vertices can reveal essential nodes that contribute significantly to the network’s connectivity or exert considerable influence. Moreover, the presence of kinks can affect processes like random walks, as high-degree vertices are more likely to attract repeated visits, thereby altering the overall dynamics of the system.
A polygon at one end of a chain, typically lacking a neighboring polygon on one of its sides, is referred to as a terminal polygon. In contrast, a polygon located within the chain, with neighboring polygons on both sides and not positioned at the chain’s ends, is termed a non-terminal polygon.
If the centers of two adjacent non-terminal polygons are not collinear, the polygon is described as kinked in the chain. There are two types of square-hexagonal kinks, denoted as
and
(
Alraqad et al., 2022). In type
, the kink is formed by a hexagon, while in type
, square occurs as a kink. A non-terminal hexagon is considered kinked if and only if it contains two consecutive vertices of degree two. Similarly, a non-terminal square is considered kinked if and only if it has a single vertex with a degree of two. Following (
Alraqad et al., 2022), we focus on square-hexagonal chains related to the kinks described below:
(1); A non-terminal hexagon that has exactly two vertices with a degree of two.
(2): A non-terminal square that is adjacent to two squares and has a vertex with a degree of two. (Figure 1A);
(3): A non-terminal square that is adjacent to a square and a hexagon and has a vertex of degree two. (Figure 1B);
(4): A non-terminal square adjacent to two hexagons and has a vertex of degree two. (Figure 1C);
FIGURE 1
In graph theory, the expected values of topological indices serve as statistical measures of a graph’s structural properties, capturing key characteristics such as connectivity, distances, and vertex degrees. These values are particularly valuable for analyzing and comparing random graphs, optimizing network designs, and predicting behaviors in fields like chemistry, biology, and social networks. Additionally, they enable the development of efficient algorithms for large-scale graph analysis by reducing the computational complexity of calculating indices across various graph models. Our work is motivated from our previous work on the kink chains introduced in (
Chunsong et al., 2024). We considered three types of kink chains,
, which are categorized as
,
, and
, based on the specific way squares and hexagons are concatenated. For generality, we divided our analysis into two cases: odd and even numbered kink chains, addressing their orders, sizes, and corresponding vertex and edge partitions. Additionally, we calculated the topological indices defined earlier and demonstrated that the second Gourava topological index is a maximizing index, while the redefined first Zagreb index is a minimizing index in both cases. Now, we will determine the expected values of topological descriptors for the newly identified kink chains
,
, and
. These kink chains are defined as;
Kink chains of.
(a) A kink chain of type in which no two adjacent vertices in the hexagons have a degree of 2, except at the terminal polygons. It is represented in Figure 2.
(b) A kink chain of type in which there are exactly two adjacent vertices with a degree of 2 in the hexagons, excluding the terminal polygons. It is represented in Figure 3.
(c) A kink chain of type in which there are three adjacent vertices with a degree of 2 in the hexagons, excluding the terminal polygons. It is represented in Figure 4.
FIGURE 2
FIGURE 3
FIGURE 4
Observe that there is no edge between two adjacent vertices of degree 2, only one edge between two adjacent vertices of degree 2, and two edges between two adjacent vertices of degree two in kink chains , and respectively, except at terminal polygons. Let represents the number of kinks (squares forming a kink). It is observed that even-numbered kink chains result when two squares are joined at the terminal, while odd-numbered kink chains occur when a hexagon is attached at the terminal. Consequently, an odd-numbered chain corresponds to a square terminal, whereas an even-numbered chain corresponds to a hexagonal terminal. These cases are mathematically represented as for odd-numbered chains and for even-numbered chains, where . The order of each kink chain is the same across these types and follows the formula . However, the size of the chain differs: for (terminal square), , and for (terminal hexagon), .
2.1 Vertex and corresponding edge partitions of , and
Let be the subclass of edge sets of , and then depends on number of kinks . Note that there are only , , , and -type of edges in each kink chain. Table 1 represents the edge partitions of each kink chain accordingly.
TABLE 1
| For | For | |||||
|---|---|---|---|---|---|---|
| 4 | 6 | |||||
| 4 | 4 | |||||
| 2 | 2 | |||||
Edge partitions of , and ; .
Let be the subclass of vertex sets of , and . depends on number of kinks . There are only vertices of degree 2, 3 and 4 in each kink chain. Table 2 represents vertex partitions of chains in both cases. Note that the vertex partitions remain same for each kink chain.
TABLE 2
| For | For | |
|---|---|---|
| , and | , and | |
Vertex partitions of , and .
3 Topological descriptors of , and
In this section, we will calculate some topological descriptors of , and using Tables 1, 2. Let denotes the kink chain for kinks of type , where p varies from 1 to 3.
Lemma 1Forthe first Revan index ofis given as;
Proof. For
For
Lemma 2For the Revan, and Redefined Zagreb, Hyper-Zagreb, and Gourava descriptors of kink chains , and are are presented in Table 3;
TABLE 3
| Topological | For | For | ||||
|---|---|---|---|---|---|---|
| descriptors | ||||||
Topological descriptors of , and ; .
4 Graphical representation of numerical values of topological descriptors of , and
In this section, we compared the above calculated topological descriptors using graphical representation of , and at different values of for odd and even numbered kink chains. From the Figures 5–11 we conclude that descriptor of , and hits a highest value for both and . It follows that is a maximizing descriptor. On the other hand descriptor of , and reaches a lowest value for both and , thus is a minimizing descriptor.
FIGURE 5
FIGURE 6
FIGURE 7
FIGURE 8
FIGURE 9
FIGURE 10
FIGURE 11
5 Expected values of topological descriptors of kink chains of type
As we know that there are only three possible kink chains (, and ) or arrangements of type , holding the conditions to make kink at each step. The kink chains for and are shown in Figure 12. For , terminal polygons are attachable in three different ways, resulting in three types , and . Considering that represents the probability of attaching terminal polygons in the first or second kind of arrangement, represents the probability of attaching the terminal polygon in the third type of arrangement. Possible arrangements of kink chains of type are shown in Figure 12.
FIGURE 12
Let the with number of kinks and probability is represented by . We compute Revan, Hyper-Zagreb, redefined Zagreb and Guorava descriptors of possible square-hexagonal kink chains . Let denotes the number of edges for with end vertices of degree i and j respectively. There are only , , , and -type of edges in . Here and are 4 and 2 implies that (Revan degrees) for and 4 respecively. From the definition, the topological descriptors can be expressed as
As is a random kink chain, it follows , , , , , and are random variables. Let us denote by , , , , , and the expected values of these descriptors respectively.
Note that if is odd in kink chains , and , then at step, even numbered kink chains are formed, and at step, odd numbered kink chains are formed. Similarly, if is even in kink chains , and at step, odd numbered kink chains are obtained and at step, even numbered kink chains are obtained again. So, we furthur divide our results in two possible stages;
(1) At stage and (2) At stage.
5.1 Results at stage
The three possible constructions at
step are as follows:
For and ; the change in edge partitions of for three possible constructions at step are shown in Tables 4, 5 respectively.
TABLE 4
| Type | no. of edges | |||||
|---|---|---|---|---|---|---|
| 1 | ||||||
| 2 | ||||||
| 3 | ||||||
Change in edge partitions of at stage for ; .
TABLE 5
| Type | no. of edges | |||||
|---|---|---|---|---|---|---|
| , 3 | +1 | |||||
Change in edge partitions of at stage for ; .
Let
be a square-hexagonal kink chain and
be the number of kinks.
(a) For ;
(b) For;
Proof. (a). When terminal polygon is a square.
Thus, we have
Using (6.1), (6.2) and (6.3), we get the following relation
Applying operator E on both sides, we get
Using recursive relation upto terms
(b). When terminal polygon is a hexagon.
Thus, we have
Using (6.4), (6.5) and (6.6), we get the following relation
Applying operator E on both sides, we get
Using recursive relation upto terms
which completes the proof.
Let
be a square-hexagonal kink chain and
be the number of kinks. Then
(a) For;
(b) For;
Thus, we have
Using (6.7), (6.8) and (6.9), we get the following relation
Using recursive relation upto terms
Thus, we have
Using (6.10), (6.11) and (6.12), we get the following relation
Using recursive relation upto terms
which completes the proof.
Let be a square-hexagonal kink chain and be the number of kinks, then
Thus, we have
Using (6.13), (6.14) and (6.15), we get the following relation
Applying operator E on both sides and
Using recursive relation upto terms
Thus, we have
Using (6.16), (6.17) and (6.18), we get the following relation
Using recursive relation upto terms
which completes the proof.
Let
be a square-hexagonal kink chain and
be the number of kinks.
(a) For ;
(b) For ;
Proof. (a). For , , which is indeed true. Let , using Table 4, we get
If
, then
If , then
If, then
Thus, we have
Using (6.19), (6.20) and (6.21), we get the following relation
Using recursive relation upto terms
Thus, we have
Using (6.22), (6.23) and (6.24), we get the following relation
Applying operator E on both sides and Using recursive relation upto terms
which completes the proof.
Let
be a square-hexagonal kink chain and
be the number of kinks.
(a) For ;
(b) For ;
Thus, we have
Using (6.25), (6.26) and (6.27), we get the following relation
Applying operator E on both sides and
Using recursive relation upto terms
Thus, we have
Using (6.28), (6.29) and (6.30), we get the following relation
Using recursive relation upto terms
Let
be a square-hexagonal kink chain and
be the number of kinks.
(a) For ;
(b) For ;
Thus, we have
Using (6.31), (6.32) and (6.33), we get the following relation
Applying operator E on both sides and
Using recursive relation upto terms
Thus, we have
Using (6.34), (6.35) and (6.36), we get the following relation
Using recursive relation upto terms
which completes the proof.
Let
be a square-hexagonal kink chain and
be the number of kinks.
(a) For ;
(b) For ;
Thus, we have
Using (6.37), (6.38) and (6.39), we get the following relation
Applying operator E on both sides and
Using recursive relation upto terms
Thus, we have
Using (6.40), (6.41) and (6.42), we get the following relation
Using recursive relation upto termswhich completes the proof.
The expected values , , , , and descriptors for ; depend on , but the is independent of for both cases. As for the sake of generality we have taken expectations in odd and even cases. Therefore, at stage the sum of expected values for and of a certain topological descriptor of three kink chains is equal to the sum of the average value of topological descriptor of three kink chain for and with a constant factor. As expectation of constant is zero, so our results are true. The values of , , , , and descriptors can be computed by using in the above proved theorems.
Corollary 1
Let
;
then at
stage;
Remark 1The value of descriptor at stage for and ; is equal and independent of , ;; , 2 and 3
It is observed that the expected values , , , , , and for ; are independent of and depends only on .
Remark 2
Let
,;
then at
stage,
5.2 Analytical expressions at stage
Now we analytically prove that at stage, for any value of and , the Gourava descriptor is always greater than the remaining six descriptors, namely, ( and ) Revan descriptors, ( and ), Zagreb descriptors, Hyper-Zagreb descriptor and Gourava descriptor for and . All the expressions holds for and for all .
Corollary 2For and ; , we have
Proof. For
which holds for and for all , so we have
For
which holds for all , so we have
Corollary 3For and ; , we have
Proof. For
which holds for and for all , so we have
For
which holds for all , so we have
Corollary 4For and ; , we have
Proof. For
which holds for and for all , so we haveForwhich holds for all , so we have
Corollary 5For and ; , we have
Proof. For
which holds for and for all , so we have
For
which holds for all , so we have
Corollary 6For and ; , we have
Proof. For
which holds for and for all , so we have
For
which holds for all , so we have
Corollary 7For and ; , we have
Proof. For
which holds for and for all , so we have
For
which holds for all , so we have
From the above analytical expressions we get;
Corollary 8
5.3 A comparison of expected values of topological descriptors at stage
Table 6, 7 depict the expected values of Revan descriptors, Zagreb descriptors, Hyper-Zagreb descriptor and Gourava descriptors for and and respectively. Observe that the value of expectation of Gourava descriptor is always greater than the remaining six descriptors in both cases.
TABLE 6
| 3 | 148.67 | 226.67 | 18 | 33.09 | 835.33 | 338 | 1,230.66 |
| 5 | 215.33 | 313.33 | 26 | 51.30 | 1,336.66 | 534 | 2013.33 |
| 7 | 282 | 400 | 34 | 69.50 | 1838 | 730 | 2,796 |
| 9 | 348.67 | 486.67 | 42 | 87.70 | 2,339.33 | 926 | 3,578.66 |
| 11 | 415.33 | 573.33 | 50 | 105.91 | 2,840.66 | 1,122 | 4,361.33 |
| 13 | 482 | 660 | 58 | 124.11 | 3,342 | 1,318 | 5,144 |
| 15 | 548.67 | 746.67 | 66 | 142.31 | 3,843.33 | 1,514 | 5,926.66 |
| 17 | 615.33 | 833.33 | 74 | 160.52 | 4,344.66 | 1710 | 6,709.33 |
| 19 | 682 | 920 | 82 | 178.72 | 4,864 | 1906 | 7,492 |
| 21 | 748.67 | 1,006.67 | 90 | 196.92 | 5,347.33 | 2,102 | 8,274.66 |
Expected values for topological descriptors at stage for and .
TABLE 7
| 4 | 178 | 296 | 22 | 34.89 | 838 | 346 | 1,216 |
| 6 | 242 | 400 | 30 | 48.22 | 1,174 | 482 | 1728 |
| 8 | 306 | 504 | 38 | 61.56 | 1,510 | 618 | 2,240 |
| 10 | 370 | 608 | 46 | 74.89 | 1846 | 754 | 2,752 |
| 12 | 434 | 712 | 54 | 88.22 | 2,182 | 890 | 3,264 |
| 14 | 498 | 816 | 62 | 101.56 | 2,518 | 1,026 | 3,776 |
| 16 | 562 | 920 | 70 | 114.89 | 2,854 | 1,162 | 4,288 |
| 18 | 626 | 1,024 | 78 | 128.22 | 3,190 | 1,298 | 4,800 |
| 20 | 690 | 1,128 | 86 | 141.56 | 3,526 | 1,434 | 5,312 |
| 22 | 754 | 1,232 | 94 | 154.89 | 3,862 | 1,570 | 5,824 |
Expected values for topological descriptors at stage for and .
5.4 Graphical representation of expected values of topological descriptors at stage
The Figures 13–16 shows that expectation of Gourava descriptor attains maximum value and of redefined Zagreb descriptor attains minimum value at stage for both the cases.
FIGURE 13
FIGURE 14
FIGURE 15
FIGURE 16
6 Applications
Kinks, which denote abrupt changes in the direction of edges within a graph, hold notable applications across diverse fields. In circuit design, minimizing kinks optimizes wire lengths and enhances efficiency. Network routing benefits from understanding kinks, as they affect data flow and network performance. Transportation planners use kink analysis to streamline traffic, plan intersections, and design efficient road networks. Graph drawing algorithms consider kinks for aesthetically pleasing and comprehensible visual representations. Lastly, in various applications where visual appeal matters, reducing kinks enhances the attractiveness and clarity of graph representations.
Studying the interforce interactions and scattering of (Lima and Almeida, 2023) kink-antikink-like solutions in a two-dimensional dilaton gravity model has practical implications in fields like material science, nonlinear optics, and cosmology. It aids in understanding energy distribution, stability dynamics, and defect interactions, which are crucial for developing advanced technologies and predicting behaviors in complex physical systems. The in situ (Zhu et al., 2023) investigations have revealed the important role for the kinks. During the growth, the creation of kinks determines the growth rate. Besides, when two domains coalesce, the shape of the final flake is affected by kinks.
In computer graphics, square and hexagonal grids are frequently used to create images or simulations. Kinks in these grids can represent corners or junctions in a digital image. Hexagonal kinks are essential in the study of tessellation and pattern generation. The expected value of random graphs plays a crucial role in graph theory as well. It helps analyze and predict various graph properties in probabilistic settings. By studying expected values in graph theory, we gain a balanced understanding of how certain structures behave under randomness, which informs both theory and practical applications. Expected values are used to calculate the probability that a random graph is connected. It can be employed to estimate the likelihood that two randomly generated graphs are isomorphic. This is valuable in assessing the structural similarity between graphs.
7 Conclusion
In this research work, we determined , , , , , and descriptors for the graphical structures of kink chains of type named as , and . We infered that is a maximizing, while is a minimizing descriptor of , and ,for both, odd and even case. Further we determined expected values of topological descriptors of at stage. We analyzed that value of descriptor of , and and is same and independent of at stage. We made numerical comparison for these expected values at stage and conclude that expected value, is greater while expected value, is smaller among other expectations , , and , , and at stage. Also, we gave exact analytical expressions of this comparison at stage which agree with numerical values of comparisons. Results at stage will be computed in the next article.
Statements
Data availability statement
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding authors.
Author contributions
RC: Conceptualization, Investigation, Methodology, Validation, Writing–review and editing. AR: Conceptualization, Investigation, Methodology, Validation, Writing–review and editing. MK: Formal Analysis, Investigation, Validation, Writing–original draft. SK: Conceptualization, Methodology, Validation, Writing–original draft. SN: Conceptualization, Formal Analysis, Investigation, Validation, Writing–review and editing. RN: Conceptualization, Formal Analysis, Investigation, Validation, Writing–review and editing.
Funding
The author(s) declare that financial support was received for the research, authorship, and/or publication of this article. This work was supported by National Natural Science Foundation of China (No. 62407011, 62172116) and the Guangzhou Academician and Expert Workstation (No. 2024-D003) and Deanship of the Science Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (KFU242800).
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Generative AI statement
The author(s) declare that no Generative AI was used in the creation of this manuscript.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
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Summary
Keywords
random square-hexagonal kink chains, topological descriptors, expected values, Randic index, Zagreb indices
Citation
Chen R, Razzaque A, Khalil M, Kanwal S, Noor S and Nazir R (2025) Expected values of topological descriptors for possible kink chains of type 2. Front. Chem. 12:1517892. doi: 10.3389/fchem.2024.1517892
Received
27 October 2024
Accepted
16 December 2024
Published
18 February 2025
Volume
12 - 2024
Edited by
Jose Luis Cabellos, Polytechnic University of Tapachula, Mexico
Reviewed by
María Guadalupe Hernández-Linares, Benemérita Universidad Autónoma de Puebla, Mexico
Xianya Geng, Anhui University of Science and Technology, China
Updates
Copyright
© 2025 Chen, Razzaque, Khalil, Kanwal, Noor and Nazir.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Asima Razzaque, arazzaque@kfu.edu.sa; Salma Kanwal, salma.kanwal@lcwu.edu.pk
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