In the realm of graph theory, recent developments have introduced novel concepts, notably the -degree and -degree, offering expedited computations compared to traditional degree-based topological indices (TIs). These TIs serve as indispensable molecular descriptors for assessing chemical compound characteristics. This manuscript aims to meticulously compute a spectrum of TIs for silicon carbide -, with a specific focus on the -degree Zagreb index, the -degree Geometric-Arithmetic index, the -degree Randić index, the -degree Atom-bond connectivity index, the -degree Harmonic index, and the -degree Sum connectivity index. This study contributes to the ongoing advancement of graph theory applications in chemical compound analysis, elucidating the nuanced structural properties inherent in silicon carbide molecules.
Semi-conductors, such as silicon, offer affordability, non-toxicity,
and find widespread utility in electronics, being integral to the
functioning of nearly all electronic devices. Silicon carbide (SiC),
composed of lightweight elements, exhibits a low thermal expansion
coefficient, strong covalent bonds, high thermal conductivity, and
remarkable hardness. Discovered by the American scientist E.G. Acheson
in 1891, this material was hailed as the hardest substance on Earth
until 1929. SiC presents various colors, such as green or black, upon
the adding of impurities like aluminum (Al), iron (Fe), or oxygen (O).
Due to its exceptional heat resistance, SiC finds application in furnace
components such as heating elements, core tubes, and refractory bricks.
Moreover, it serves as a precursor for graphene sheets [1, 2]. Its versatile properties
contribute to its extensive usage in electronics, transportation
vehicles, and applications in quantum physics. For further insights,
refer to [3, 4, 5]. This paper
delves into the topological properties of silicon carbide –.
Chemical graph theory is a field of discrete mathematics that
addresses various chemical challenges. It involves the exploration of
chemical structures present in molecular compounds relevant to
pharmaceuticals and artificial food products [5, 6, 7, 8, 9, 10]. The interdisciplinary nature of
chemical graph theory lies in its connection between chemistry and
mathematics. Notably, graph theory was pioneered by Euler in the 18th
century [11].
In chemical graph theory, molecules are typically represented as
simple connected graphs, with chemical bonds depicted as edges and atoms
as vertices. This graphical representation enables scientists to
investigate and comprehend isomerism phenomena in chemical compounds. By
studying the graph structures, researchers can analyze the behavior of
different isomers of the same chemical compound.
Furthermore, chemical graph theory finds applications in the
detection and resolution of drug-related issues [12, 13, 14, 15].
The numerical value associated with a molecular graph is known as a
topological index, which represents a unique type of graph invariant.
These molecular descriptors play a significant role in Quantitative
Structure-Activity Relationship (QSAR) studies [16]. A topological index can be conceptualized as a
function that assigns each molecular structure a real number. One of the
earliest topological indices introduced is the Wiener index, proposed by
H. Wiener in 1947 [17].
Topological indices serve as valuable tools for predicting the
physicochemical properties and bioactivity of chemical compounds. Over
the years, hundreds of topological descriptors have been defined to
better understand the structural characteristics of these compounds
[18].
The concept of –degree and –degree based
Topological Indices (TIs) in graph theory was proposed by Chellali et
al. [19]. Subsequently,
Horoldagva et al. [20]
extended these indices to mathematics. The –degree and –degree based
Zagreb and Randić indices are considered more powerful than classical
vertex-type indices. For more detailed information about –degree and –degree based TIs,
refer to [21, 22, 23]. Zhong
[24] introduced the harmonic
index, while Randić defined the Randić index in 1975 [25], and Gutman introduced the
first and second Zagreb indices [26]. Initially, these indices were based on
classical degrees, but the -degree and -degree versions of these TIs
offer more benefits. For more advanced information about graphs, silicon
carbide, -degree and
-degree, and topological
indices, see [27, 28, 29, 30, 31].
2. Basic Definitions and
Notations
Let be an
undirected, connected, and simple graph, where denotes the collection of edges
and denotes the collection
of nodes. A simple graph is one that does not have a loop or multiple
edges. If a graph has a connection between any two nodes, it is said to
be connected. Silicon carbide’s 2D molecular structures are both simple
and interconnected. The degree of a vertex , denoted as , is the number of
vertices connected to a fixed vertex . An edge is represented by .
2.1. Definitions
If is a simple connected
graph, the degree () represents the
count of different edges incident to any node within the closed
neighborhood of . The
vertex-edge degree () can be
calculated by considering the number of distinct edges incident on any
node within its open
neighborhood. Moreover, the edge-vertex degree () of an edge
is defined as the count
of vertex unions between the open neighborhoods of the endpoints and . The and based
Topological Indices (TIs) are presented below in mathematical
notation.
The degree-based
Zagreb index can be determined as follows: The 1–degree Zagreb
alpha index ())
is determined as: The 1–degree Zagreb
beta index ())
computed as: The second –degree Zagreb
index ()) mathematically defined as:
The –degree Randić
index (R()) mathematically satiated as:
The –degree
Randić index (()) determined as: The –degree Atom Bond
Connectivity index (ABC()) calculated by formula given
below, as: The –degree Geometric
Arithmetic index (GA()) determined as: The
–degree
Harmonic index (H()): The –degree
Sum-Connectivity index (X()) computed as:
Yamac and Cancan discuss this and degree based TIs for the
Sierpinski Gasket Fractal in 2009 [27].
3. Techniques
We utilized a diverse array of methodologies to obtain our findings,
encompassing the edge parcel technique, vertex segment strategy, graph
hypothetical device, degree verification tactic, and combinatorial
techniques. In this investigation, we applied various tools and
methodologies. For computational tasks and verification processes, MATLAB was employed, while MAPLE was utilized for generating 2D and 3D
graphs. Additionally, chem-sketch
software was employed for constructing structural graphs of .
4. 2D Structure of
Silicon Carbide
The molecular graph of – is shown in Figure . Any chemical compound’s building block
is the unit cell, as we all know. A molecular structure is made up of a
huge number of unit cells arranged in a certain pattern.In a molecular
structure, r represents the number of unit cells in a row, where s
represents the number of rows. In Figure a unit cell and a structure of and , and and are represented. Consequently, the
total numbers of vertices, edges and faces in are;
FIgure 1: 2-Dimensional Structure of -, (A) Chemical Unit Cell of - (B) -, (C) - (D) -, Where Silicon Atoms Si are Blue and Carbon Atoms C Are Brown
5. Methodology
of Silicon Carbide
Formulas
The unit cell is used to compute silicon carbide formulae –. To raise , interconnect the unit cells
horizontally, then connect the rows vertically to increase . The connection points must be correct.
Where is the number of rows and
is the number of columns.
5.1. Vertex Partition
There are kinds of nodes based
on the degree of nodes. Vertices of 1, 2 and 3 degree are represented as , and respectively as shown in Table 1.
Table 1: Vertex Partition of –
3
6
9
3
6
9
3
6
9
4
6
8
8
10
12
12
14
16
3
8
13
9
24
39
15
40
65
Table 2: Degree of Vertex with Corresponding Cardinality
deg()
Cardinality
Table 3: Vertex and Edges Frequency of –
Total vertices
Total edges
5.2. Edge Partition
By using above methodology we will partition the edges of –. In the instance of –, there are five distinct edge
portions, as shown in Table 4. It is important to note that the variables
.
Table 4: –Degree of –
–degree
Cardinality
6. Main Results For
Silicon Carbide –
In this section, we calculate the main results for silicon carbide
–. We calculate the TIs using
different basic definitions and values given in tables. The specific TI
index uses specific values in the table and provides information about
the correlation coefficient. These correlation constants represent the
connection between the numerical number and the characterization of any
graph or network.
Table 5: –Degree of
– for all
–degree
Cardinality
1
2
2
1
3
2
4
2
2
5
3
6
2
3
7
3
8
3
9
Table 6: –Degree of End
Vertices of Each Edge of –
deg(),deg()
–degree
Cardinality
(3,1)
(7,3)
(2,1)
(4,2)
2
(2,2)
(5,5)
(2,2)
(7,4)
1
(3,2)
(7,5)
3
(3,2)
(8,4)
1
(3,2)
(8,5)
(3,3)
(8,7)
(3,3)
(8,8)
(3,3)
(9,7)
(3,3)
(9,8)
(3,3)
(9,9)
(i) The index:
By making use of –degree of edge
partitions of –, as shown in Table 5, we calculate the
index in the following
lines:
(ii) The
index:
By making use of –degree of
vertices partition of – for , as seen in Table 5, we compute the
in the following
lines:
(iii) The
index:
By making use of –degree based
partition of the end vertices of the edges of – for , as shown in Table 6, we compute the
in the following
lines:
(iv) The index:
Simply availing use of –degree based
partition of end vertices of the edges of – for , using Table 6, we compute the
in the following lines:
(v) The index:
By utilizing the –degree based
partition of end vertices of the edges of – for , as given in Table 6, we compute the
as follows,
(vi) The index:
By utilizing the –degree of edges
partition of – for , as given in Table 5, we compute the
as follows,
(vii) The index:
With the help of –degree based
partition of the end vertices of the edges of – for , as shown in Table 6, we compute the
as follows,
(viii) The index:
By making use of –degree based
partition of the end vertices of the edges of – for , as shown in Table 6, we compute the
as follows,
(ix) The index:
By making use of –degree based
partition of the end vertices of the edges of – for , as shown in Table 6, we compute the
as follows,
(x) The index:
By making use of –degree based
partition of the end vertices of the edges of – for , as shown in Table 6, we compute the
as follows,
7. Applications
The topological indices provide an easy way to convert chemical
composition into numerical values that can be correlated with physical
characteristics in QSPR research. The ev and ve-related indices give
more effective results as compared to classical indices in various
cases. For instance, the correlation coefficient between the acentric
factor of 18 isomers of octane and the classical first Zagreb index is
moderate (), but the
ev-degree Zagreb index shows excellent values of correlation as . Similarly, the correlation
between properties of octane (acentric factor, boiling point, and
entropy) and classical R-index is very low, as 4, , and , but ev-degree-related
Randić indices give amazing values of coefficients like , , and . The H-index also shows
moderate correlation values with the characteristics of 18 isomers of
octane: r(AF) = 0.7998, r(entropy) = 0.7594, and r(BP) = 0.801. This
ev-degree type approach is also effective in discussing the structural
features of the alkane family and SiC isomers.
8. Conclusion
Application of silicone carbide in physical perspective, deoxidizer
used in steel making, one of the most widely used refractory materials
with the best economic benefits, high-quality abrasive for sandblasting.
Furthermore, because of small amounts of iron or different contaminating
influences from the current generation, this is typically discovered as
a somewhat blue dark, brilliant crystalline strong. In this study, we
defined topological invariants of silicon carbide – based on –degree and –degree. The
findings are extremely valuable and beneficial from both a chemical and
pharmacological standpoint. In future, we can find –degree and –degree
topological indices of some nanostar dendrimers.
Conflict of
Interest
The authors declare no conflict of interest.
Funding
Malkesh Singh is receiving funds from SMVDU vide Grant/Award Number:
SMVDU/R&D/21/5121-5125 for this research.
References:
Novoselov, K. S., McCann, E., Morozov, S. V.,
Fal’ko, V. I., Katsnelson, M. I., Zeitler, U., Jiang, D., Schedin, F.
and Geim, A. K., 2006. Unconventional Quantum Hall Effect and Berry’s
Phase of 2p in Bilayer Graphene. Nature Physics, 2(3),
pp.177-188.
Zhong, Y., Shaw, L. L., Manjarres, M. and Zawrah, M. F.,
2010. Synthesis of Silicon Carbide Nanopowder Using Silica Fume.
Journal of the American Ceramic Society, 93(10), pp.3159-3167.
Zetterling, C. M., 2012. Present and future applications of Silicon
Carbide devices and circuits. In Proceedings of the IEEE 2012 Custom
Integrated Circuits Conference (pp. 1-8). IEEE.
O’Sullivan, D., Pomeroy, M. J., Hampshire, S. and Murtagh, M. J.,
2004. Degradation resistance of silicon carbide diesel particulate
filters to diesel fuel ash deposits. MRS Proceedings, 19(10),
pp.2913-2921.
Ruddy, F. H., Ottaviani, L., Lyoussi, A., Destouches, C., Palais, O.
and Reynard-Carette, C., 2021. Performance and Applications of Silicon
Carbide Neutron Detectors in Harsh Nuclear Environments. In EPJ Web
of Conferences, 253, p. 11003. EDP Sciences.
Trinajstic, N., 1992. Chemical Graph Theory. CRC Press.
Costa, P., Evangelista, J. S., Leal, I. and Miranda, P. C., 2021.
Chemical Graph Theory for Property Modeling in QSAR and QSPR—Charming
QSAR & QSPR. Mathematics, 9(1), p.60.
Garcia-Domenech, R., Galvez, J., de Julian-Ortiz, J. V. and Pogliani,
L., 2008. Some new trends in chemical graph theory. Chemical
Reviews, 108(3), pp.1127-1169.
Estrada, E., 2008. Quantum-Chemical Foundations of the Topological
Sub-Structural Molecular Design. Journal of Physical Chemistry A,
112(21), pp.5208-5217.
Konstantinova, E. V., 2006. On some applications of information
indices in chemical graph theory. In General Theory of Information
Transfer and Combinatorics, pp. 831-852. Springer, Berlin,
Heidelberg.
Alexanderson, G., 2006. About the cover: Euler and Königsberg’s
Bridges: A historical view. Bulletin of the American Mathematical
Society, 43(4), pp.567-573.
Park, S., Yuan, Y. and Choe, Y., 2021. Application of graph theory to
mining the similarity of travel trajectories. Tourism Management,
87, p.104391.
Carbó-Dorca, R., Robert, D., Amat, L., Gironés, X. and Besalú, E.,
2000. Molecular Quantum Similarity in QSAR and drug Design (Vol.
73). Springer Science & Business Media.
Toman, J. and Olszewska, J. I., 2014. Algorithm for graph building
based on Google Maps and Google Earth. In 2014 IEEE 15th
International Symposium on Computational Intelligence and Informatics
(CINTI), pp. 55-60. IEEE.
Deo, N., 2017. Graph theory with applications to engineering and
computer science. Courier Dover Publications.
Sigarreta, J. M., 2021. Mathematical Properties of Variable
Topological Indices. Symmetry, 13(1), p.43.
Wiener, H., 1947. Structural determination of paraffin boiling
points. Journal of the American Chemical Society, 1(69),
pp.17-20.
Mondal, S., De, N. and Pal, A., 2021. Multiplicative degree based
topological indices of nanostar dendrimers. Biointerface research in
applied chemistry, 11(1), pp.7700-7711.
Chellali, M., Haynes, T. W., Hedetniemi, S. T. and Lewis, T. M.,
2017. On ve-Degrees and ev-Degrees in Graphs. Discrete Mathematics,
340(2), pp.31-38.
Horoldagva, B., Das, K. C. and Selenge, T. A., 2019. On ve-Degree and
ev-Degree of Graphs. Discrete Optimization, 31, pp.1-7.
Sahin, B. and Ediz, S., 2018. On ev-Degree and ve-Degree Topological
Indices. Iranian Journal of Mathematical Chemistry, 9(4),
pp.263-277.
Cancan, M., 2019. On Ev-Degree and Ve-Degree Topological Properties
of Tickysim Spiking Neural Network. Computational Intelligence and
Neuroscience, 2019, pp.1-7.
Cai, Z.Q., Rauf, A., Ishtiaq, M. and Siddiqui, M.K., 2022. On
ve-degree and ev-degree based topological properties of silicon carbide
Si2C3-II [p, q]. Polycyclic Aromatic Compounds, 42(2),
pp.593-607.
Zhong, L., 2012. The Harmonic Index for Graphs. Applied
Mathematics Letters, 25(3), pp.561-566.
Randić, M., 1975. On characterization of molecular branching.
Journal of the American Chemical Society, 97(1975),
pp.6609-6615.
Xu, K., Das, K. C. and Balachandran, S., 2014. Maximizing the Zagreb
indices of (n, m)-graphs. MATCH Communications in Mathematical and
in Computer Chemistry, 72, pp.641-654.
Yamaç, K. and Cancan, M., 2019. On ev-degree and ve-degree based
topological properties of the sierpinski gasket fractal. Sigma,
37(4), pp.1275-1280.
Hu, M., Ali, H., Binyamin, M. A., Ali, B., Liu, J.-B. and Fan, C.,
2021. On Distance-based Topological Descriptors of Chemical
Interconnection Networks. Journal of Mathematics, 2021,
p.5520619, 10 Pages.
Huo, Y., Ali, H., Binyamin, M. A., Asghar, S. S., Babar, U. and Liu,
J.-B, 2020. On Topological Indices of mth Chain Hex-Derived Networks of
the Third Type. Frontiers in Physics, 8, p.593275.
Zhao, X., Ali, H., Ali, B., Binyamin, M. A., Liu, J.-B. and Raza, A.,
2021. Statistics and Calculation of Entropy of Dominating David Derived
Networks. Complexity, 2021, pp.1-15.
Azeem, M., Aslam, A., Iqbal, Z., Binyamin, M. A. and Gao, W. (2021).
Topological aspects of 2D structures of trans-Pd(NH2)S lattice and a
Metal-organic Superlattice. Arabian Journal of Chemistry,
14(3), p.102963.