Advertisement
Browse Subject Areas
?

Click through the PLOS taxonomy to find articles in your field.

For more information about PLOS Subject Areas, click here.

  • Loading metrics

Open Access

Peer-reviewed

Research Article

Valency-based structural properties of gamma-sheet of boron clusters

  • Ali N. A. Koam,

    Roles Conceptualization, Data curation, Formal analysis, Funding acquisition, Writing – original draft, Writing – review & editing

    Affiliation Department of Mathematics, College of Science, Jazan University, New Campus, Jazan, Saudi Arabia

    ⨯
  • Muhammad Azeem ,

    Roles Investigation, Methodology, Project administration, Resources, Writing – original draft, Writing – review & editing

    azeemali7009@gmail.com

    Affiliation Department of Mathematics, Riphah International University, Lahore, Pakistan

    ⨯
  • Ali Ahmad

    Roles Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review & editing

    Affiliation College of Computer Science & Information Technology, Jazan University, Jazan, Saudi Arabia

    ⨯

Abstract

Boron cluster sheets are two-dimensional boron atom-based formations called borophene. They are similar to the two-dimensional sheet known as graphene, which is composed of carbon atoms arranged in a hexagonal lattice. The unique electrical, mechanical, and thermal properties of borophene make it a sought-after substance for a variety of uses, such as catalysis, energy storage, and electronics. There are two ways to manufacture borophene: chemical vapor deposition and molecular beam epitaxy. Vertex-edge valency-based topological descriptors are a great example of a molecular descriptor that provides information on the connection of atoms in a molecule. These descriptions are based on the notion that a node’s value in a molecular network is the sum of the valency of those atoms that are directly connected to that node. In this article, we discussed some novel vertex-edge (ve) and edge-vertex (ev) topological descriptors and found their formulations for the boron cluster or borophene sheets. Also, we show the numerical and graphical comparison of these descriptors in this article.

Citation: Koam ANA, Azeem M, Ahmad A (2024) Valency-based structural properties of gamma-sheet of boron clusters. PLoS ONE 19(5): e0303570. https://doi.org/10.1371/journal.pone.0303570

Editor: Mingsen Deng, Guizhou University of Finance and Economics, CHINA

Received: March 4, 2024; Accepted: April 26, 2024; Published: May 23, 2024

Copyright: © 2024 Koam et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All data are in the manuscript, there is no external data available for this submission.

Funding: The author(s) received no specific funding for this work;.

Competing interests: The authors declare that they have no conflicts of interest.

1 Introduction

A fundamental study in the structural theory of chemical graphs is a molecular structure diagram, in which atoms are seen as atoms and lines as chemical bonds. If there is a path connecting any two atoms in a diagram, it is said to be linked. A system is a linked map with a loop and no other lines separating any two atoms. The number of atoms is linked to a given node. The valency of v is indicated by the symbol dv. The open neighbourhood of node v is the collection of all the atoms that are close to it, and it may be represented by the symbol N(v). When the node v, indicated by N[v], was included, the open neighbourhood changed to the closed neighbourhood.

The total length of the most constrained path between two atoms is the distance across both. In physical science, the concepts of valence and valency are pretty closely related. For basic diagram invariants, see [1]. The qualities and unstudied material’s natural exercises are predicted by the relationship between the QSPR and QSAR. Topological descriptors and a few physicochemical features are used in such compounds for estimating the bioactivity of chemical substances [2–5]. In a diagram of a chemical compound, a number is used to express a topological descriptor that may be used to depict the highlighted chemical molecule and predict its physiochemical characteristics.

Wiener devised the topological descriptor structure in 1947. He presented the Wiener descriptor and was given an approximation of the alkanes’ point of boiling [6–10]. More than 3000 topological indicators have been characterized so far, however, no one descriptor is enough to determine all physicochemical characteristics, yet these topological descriptors together can partly do this. After that the Randić descriptor was given by cite7 in 1975. One of the most significant, widely used, and applicable topological descriptors is the Randić descriptor. Graph invariance is the subject of several surveys, studies, and books [11–16]. For further detail, we refer to see [17–22].

Authors of [23], first proposed two unique valency concepts, the ve-valency and the ev-valency, and [24, 25] made a contribution to the research on the “ve-valency” and the “ev-valency”. When current descriptors were combined with the newly developed valency-based descriptors, the outcomes were improved, as shown in [26–28]. It is currently shown that the ve-valency Zagreb descriptor has a stronger approximation performance than the original Zagreb descriptor.

There are maany more graph theoretical parameters other than topological descriptors like resolvability parameters [29–42].

2 The ve-valency and ev-valency based topological descriptors

The definition of an edge’s ev-valency was provided by the contributors of citation number [23], and it is the count of atoms in the union of the closed neighbourhoods of u and v, where e = uv ∈ E which is denoted by dev(e). The count of routes of various lines that are incident to any node from the closed neighbourhood of v is known as the ve-valency of the node v ∈ V, indicated by the symbol dve(v). For the sake of this essay, we will assume that G is a connected graph with e = uv ∈ E(G) while v ∈ V. In this section, we provide some fundamental notions of ve-valency and ev-valency topological descriptors, [23, 43]. Some ev-valency related topological descriptors are: the ev-valency Randić descriptor, ansd ev-valency Zagreb descriptor, and topological descriptors related to ev-valency are: The first ve-valency Zagreb α descriptor, ve-valency sum-connectivity descriptor, ve-valency harmonic descriptor, ve-valency geometric-arithmetic descriptor, ve-valency atom-bond connectivity descriptor, ve-valency Randić descriptor, second ve-valency Zagreb descriptor, and first ve-valency Zagreb β descriptor. All these descriptors for any graph G can be found in [44], are formulated as:

  • The ev-valency Zagreb Index:
  • The first ve-valency Zagreb α descriptor:
  • The first ve-valency Zagreb β descriptor:
  • The second ve-valency Zagreb descriptor:
  • The ve-valency Randić descriptor:
  • The ev-valency Randić descriptor:
  • The atom-bond connectivity descriptor:
  • The geometric-arithmetic descriptor:
  • The harmonic descriptor:
  • The sum-connectivity descriptor:

3 The γ−sheet of boron clusters

Boron cluster sheets are two-dimensional boron-atom-based formations called borophene. They are similar to the two-dimensional sheet known as graphene, which is composed of carbon atoms arranged in a hexagonal lattice. The unique electrical, mechanical, and thermal properties of borophene make it a sought-after substance for a variety of uses, such as catalysis, energy storage, and electronics. There are two ways to manufacture borophene: chemical vapour deposition and molecular beam epitaxy. To create borophene sheets, many arrangements of boron atoms, such as triangular, honeycomb, and rectangular patterns, may be created. Fig 1 shows the a graph of γ−sheet of boron clusters for the particular values of p = 6 and q = 3. We established our computational findings in this part for γ−sheet of boron clusters (it is denoted by γ(p, q)). The count of atoms and edges in γ(p, q) are 5pq + 2q and 12pq − p + q respectively, shown in the Table 1.

thumbnail
Download:
Fig 1. γ−sheet of Boron Clusters for p = 6 and q = 3.

https://doi.org/10.1371/journal.pone.0303570.g001

thumbnail
Download:
Table 1. Numerical comparison of , and Rev.

https://doi.org/10.1371/journal.pone.0303570.t001

In Table 2, we partitioned the lines based on ev-valency of the γ(p, q). In Table 3, we partition the atoms based on ve-valency of γ(p, q). We partition the lines based on ve-valency of the end atoms of γ(p, q) in Table 4.

thumbnail
Download:
Table 2. Line partition of γ(p, q).

https://doi.org/10.1371/journal.pone.0303570.t002

thumbnail
Download:
Table 3. Node partition of γ(p, q).

https://doi.org/10.1371/journal.pone.0303570.t003

thumbnail
Download:
Table 4. The line partition of ve-valency based of its end atoms of γ(p, q).

https://doi.org/10.1371/journal.pone.0303570.t004

4 Main results

In this part, we will determine different kinds of descriptors’ ev-valency and ve-valency based descriptors, which are listed as follows;

  • ev−valency Zagreb Index.

Using the statistics given in Table 2, we determine the ev-valency based Zagreb descriptor of γ(p, q) as:

  • The first ve-valency Zagreb α descriptor.

Using the statistics given in Table 3, we determine the ve-valency based Zagreb descriptor of γ(p, q) as:

  • The first ve-valency Zagreb β descriptor.

Using the statistics given in Table 4, we determine the ve-valency based Zagreb descriptor of γ(p, q) as:

  • The second ve-valency Zagreb descriptor.

Using the statistics given in Table 4, we determine the ve-valency based Zagreb descriptor of γ(p, q) as:

  • The ve-valency Randić descriptor.

Using the statistics given in Table 4, we determine the ve-valency based Randić descriptor of γ(p, q) as:

  • The ev-valency Randić descriptor.

Using the statistics given in Table 2, we determine the ev-valency based Randić descriptor of γ(p, q) as:

  • The atom-bond connectivity descriptor.

Using the statistics given in Table 4, we determine the ve-valency based atom-bond connectivity descriptor of γ(p, q) as:

  • The geometric-arithmetic descriptor.

Using the statistics given in Table 4, we determine the ve-valency based geometric arithmetic descriptor of γ(p, q) as:

  • The harmonic descriptor.

Using the statistics given in Table 4, we determine the ve-valency based Harmonic descriptor of γ(p, q) as:

  • The sum-connectivity descriptor.

Using the statistics given in Table 4, we determine the ve-valency based sum connectivity descriptor of γ(p, q) as:

5 Numerical and graphical representation and discussion

Both numerical and graphical calculations are used to determine the ve and ev for ten distinct categories of valency-based topological descriptors for the γ(p, q). The behavior of the first Zagreb alpha descriptor, first Zagreb beta descriptor, and second Zagreb descriptor is practically identical in the growing orientation as the quantity of n rises, while the appreciate of the ev Zagreb descriptor improves very quickly with the increased value of n, as can be seen in Fig 2. The patterns of behavior of the atom bond connectivity descriptor and geometric arithmetic descriptor are hardly noticeably growing with an elevated value of n, but the value of the ev Randić descriptor has a very quick rise with the increased value of n, as can be seen in Fig 3.

thumbnail
Download:
Fig 2. Graphical Comparison of , and Rev.

https://doi.org/10.1371/journal.pone.0303570.g002

thumbnail
Download:
Fig 3. Graphical Comparison of ABCve, GAve, Rve, Hve and χve.

https://doi.org/10.1371/journal.pone.0303570.g003

The numerical and graphical representations of γ(p, q) are shown below.

6 Closing remarks

Topological indicators have several uses in fields such as chemical graph theory, computer science, networks, agriculture, etc. These characteristics aid in determining how their frameworks behave. We generated 10 distinct sorts of topological descriptors based on ev and ve valency for the γ-sheet of boron clusters. Their numerical values for different amounts of n have been computed after their clear formulations have been determined. In addition, we compared the charts and spoke about their behavior. We note that when the quantity of n grows, all descriptor values rise, results are shown in the Tables 5 and 6.

thumbnail
Download:
Table 5. Numerical comparison of , and Rev.

https://doi.org/10.1371/journal.pone.0303570.t005

thumbnail
Download:
Table 6. Numerical comparison of ABCve, GAve, Rve, Hve and χve.

https://doi.org/10.1371/journal.pone.0303570.t006

6.1 Future directions and open problems

Researchers working in topological numbers of graphs will find guidance and direction in this section. There are many topological numbers, like connection number based topological index is one the next work can be done after this chosen topic.

6.2 Limitations of the proposed model

There are few limitations of this work, like Computational Complexity: Certain topological indices or computational models may exhibit significant computational complexity, rendering them unfeasible for handling extensive datasets or real-time applications.

References

  1. 1. West, D. B. (2001). Introduction to graph theory (Vol. 2). Upper Saddle River: Prentice hall.
    • 2. Rcker G. and Rcker C. (1999). On topological descriptors, boiling points, and cycloalkanes. Journal of chemical information and computer sciences, 39(5), 788–802.
    • 3. Klavzar S., and Gutman I. (1996). A comparison of the Schultz molecular topological descriptor with the Wiener descriptor. Journal of chemical information and computer sciences, 36(5), 1001–1003.
    • 4. Bruckler F. M., Doslic T., Graovac A., and Gutman I. (2011). On a class of distance-based molecular structure descriptors. Chemical physics letters, 503(4-6), 336–338.
    • 5. Deng H., Yang J., and Xia F. (2011). A general modeling of some vertex-valency based topological descriptors in benzenoid systems and phenylenes. Computers and Mathematics with Applications, 61(10), 3017–3023.
    • 6. Wiener H. (1947). Structural determination of paraffin boiling points. Journal of the American Chemical Society, 69(1), 17–20. pmid:20291038
    • 7. Randić M. (1975). Characterization of molecular branching. Journal of the American Chemical Society, 97(23), 6609–6615.
    • 8. Bollobas B., and Erdos P. (1998). Graphs of extremal weights. Ars Combinatoria, 50, 225–233.
    • 9. Amic D., Beslo D., Lucic B., Nikolic S., and Trinajstic N. (1998). The vertex-connectivity descriptor revisited. Journal of chemical information and computer sciences, 38(5), 819–822.
    • 10. Hu Y., Li X., Shi Y., Xu T., and Gutman I. (2005). On molecular graphs with smallest and greatest zeroth-order general Randic descriptor. MATCH Commun. Math. Comput. Chem, 54(2), 425–434.
    • 11. Li, X., Gutman, I., and Randić, M. (2006). Mathematical aspects of Randić-type molecular structure descriptors. University, Faculty of Science.
      • 12. Randić M. (2008). On history of the Randić descriptor and emerging hostility toward chemical graph theory. MATCH Commun. Math. Comput. Chem, 59, 5–124.
      • 13. Randić M. (2001). The connectivity descriptor 25 years after. Journal of Molecular Graphics and Modelling, 20(1), 19–35. pmid:11760000
      • 14. Gutman, I., and Furtula, B. (Eds.). (2008). Recent results in the theory of Randic descriptor. University, Faculty of Science.
        • 15. Li X., and Shi Y. (2008). A survey on the Randic descriptor. MATCH Commun. Math. Comput. Chem, 59(1), 127–156.
        • 16. Nikolic S., Kovacevic G., Milicevic A., and Trinajstic N. (2003). The Zagreb descriptors 30 years after. Croatica chemica acta, 76(2), 113–124.
        • 17. Gutman I., and Das K. C. (2004). The first Zagreb descriptor 30 years after. MATCH Commun. Math. Comput. Chem, 50(1), 83–92.
        • 18. Das K. C., and Gutman I. (2004). Some properties of the second Zagreb descriptor. MATCH Commun. Math. Comput. Chem, 52(1), 3–1.
        • 19. Gutman I., Milicevic A., Nikolic S., and Trinajstic N. (2010). About the Zagreb Indices. Kemija u Industriji, 59(12), 577–589.
        • 20. Vukicevic D., and Graovac A. (2004). Valence connectivity versus Randić, Zagreb and modified Zagreb descriptor: A linear algorithm to check discriminative properties of descriptors in acyclic molecular graphs. Croatica chemica acta, 77(3), 501–508.
        • 21. Milicevic A., Nikolic S., and Trinajstic N. (2004). On reformulated Zagreb descriptors. Molec- ular diversity, 8(4), 393–399.
        • 22. Shigehalli V. S., and Kanabur R. (2016). Computation of new valency-based topological descriptors of graphene. Journal of Mathematics, 2016.
        • 23. Chellali M., Haynes T.W., Hedetniemi S.T. and Lewis T.M., 2017. On ve-valencys and ev-valencys in graphs. Discrete Mathematics, 340(2), pp.31–38.
        • 24. Horoldagva B., Das K.C. and Selenge T.A., 2019. On ve-valency and ev-valency of graphs. Discrete Optimization, 31, pp.1–7.
        • 25. Wang N., Li X., Lian X., Zhuang Q., Wang J., Li J., et al. (2024). Acetate Ions Facilitated Immobilization of Highly Dispersed Transition Metal Oxide Nanoclusters in Mesoporous Silica. Inorganic Chemistry, 63(9), 4393–4403. pmid:38375640
        • 26. Ediz S., 2017. A new tool for QSPR researches: ev-valency Randi descriptor. Celal Bayar University Journal of Science, 13(3), pp.615–618.
        • 27. Sahin B. and Ediz S., 2018. On ev-valency and ve-valency topological descriptors. Iranian Journal of Mathe- matical Chemistry, 9(4), pp.263–277.
        • 28. Ediz S., 2018. On ve-valency molecular topological properties of silicate and oxygen networks. International Journal of Computing Science and Mathematics, 9(1), pp.1–12.
        • 29. Khabyah A. A., Jamil M. K., Koam A. N. A., Javed A., Azeem M., “Partition dimension of COVID antiviral drug structures”, Mathematical Biosciences and Engineering, 19(10), 10078–10095, 2022.
        • 30. Azeem M., Jamil M. K., Javed A., Ahmad A., “Verification of some topological indices of Y-junction based nanostructures by M-polynomials”, Journal of Mathematics, June 2022.
        • 31. Azeem M., Imran M., Nadeem M. F., “Sharp bounds on partition dimension of hexagonal Möbius ladder”, Journal of King Saud University-Science, Dec. 2021.
        • 32. Azeem M., Anwar S., Jamil M. K., Saeed M., Deveci M., Topological Numbers of Fuzzy Soft Graphs and Their Application, Information Sciences, Jan. 2024.
        • 33. Anwar S., Azeem M., Jamil M. K., Topological numbers of fuzzy soft graphs and their applications in globalizing the world by mutual trade, Applied Soft Computing Journal, April. 2024.
        • 34. Azeem M., Nadeem M. F., “Metric-based resolvability of polycyclic aromatic hydrocarbons”, The European Physical Journal Plus, 136(4), 1–14, 2021.
        • 35. Bukhari S., Jamil M. K., Azeem M., Swaray S., Honeycomb Rhombic Torus Vertex-Edge Based Resolvability Parameters and its Application in Robot Navigation, IEEE Access, Jan. 2024.
        • 36. Azeem M., Cycle-Super Magic Labeling of Polyomino Linear and Zig-Zag Chains, Journal of Operations Intelligence, Dec. 2023.
        • 37. Shanmukha M. C., Ismail R., Gowtham K. J., Usha A., Azeem M., Al-Sabri E. H. A. Chemical applicability and computation of K-Banhatti indices for benzenoid hydrocarbons and triazine-based covalent organic frameworks, Scientific Reports, Oct. 2023. pmid:37853135
        • 38. Azeem M., Jamil M. K., Shang Y., Notes on the Localization of Generalized Hexagonal Cellular Networks, Mathematics, Feb 2023.
        • 39. Bukhari S., Jamil M. K., Azeem M., Swaray S., Patched Network and its Vertex-Edge Metric-Based Dimension, IEEE Access, Jan 2023.
        • 40. Shao Z., Wu P., Gao Y., Gutman I., Zhang X., On the maximum ABC index of graphs without pendent vertices, Applied Mathematics and Computation, Volume 315, 15 December 2017, Pages 298–312.
        • 41. Ediz S., On novel topological characteristics of graphene. Physica Scripta, 98(11), 115220, 2023.
        • 42. Ediz S., Cancan M., Alaeiyan M., Farahani M. R., Ve-degree and Ev-degree topological analysis of some anticancer drugs. Eurasian Chemical Communications, 2(8), 834–840, 2020.
        • 43. Cancan M., 2019 On ev-Degree and ve-Degree Topological Properties of Tickysim Spiking Neural Network. Computational Intelligence and Neuroscience, 2019. pmid:31281340
        • 44. Koam A. N. A., Azeem M., Ahmad A., Masmali I., Connection Number-based Molecular Descriptors of Skin Cancer Drugs, Ain Shams Engineering Journal, March. 2024.