On Ve-Degree and Ev-Degree Based Topological Invariants of Chemical Structures

Niat Nigar1, Sajid Mahboob Alam1, Muhammad Waheed Rasheed2, Mohammad Reza Farahani3, Mehdi Alaeiyan3, Murat Cancan4
1Department of Mathematics, Minhaj University, Lahore, Pakistan
2Department of Mathematics, Division of Science and Technology, University of Education, Lahore, Pakistan
3Department of Mathematics and Computer Science, Iran University of Science and Technology(IUST), Narmak, Tehran, 16844, Iran
4Faculty of Education, Van Yuzuncu Yl University, Zeve Campus, Tuba, 65080, Van, Turkey

Abstract

In the realm of graph theory, recent developments have introduced novel concepts, notably the νε-degree and εν-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 SiC4-I[r,s], with a specific focus on the εν-degree Zagreb index, the νε-degree Geometric-Arithmetic index, the εν-degree Randić index, the νε-degree Atom-bond connectivity index, the νε-degree Harmonic index, and the νε-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.

Keywords: Graphs, Ev-degree, Topological indices, Ve-degree, Silicon Carbide

1. Introduction

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 SiC4I[r,s].

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 νεdegree and ενdegree 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 νεdegree and ενdegree based Zagreb and Randić indices are considered more powerful than classical vertex-type indices. For more detailed information about ενdegree and νεdegree 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 ϵν-degree and νϵ-degree versions of these TIs offer more benefits. For more advanced information about graphs, silicon carbide, ϵν-degree and νϵ-degree, and topological indices, see [27, 28, 29, 30, 31].

2. Basic Definitions and Notations

Let ζ)=(V,E) be an undirected, connected, and simple graph, where E(ζ) denotes the collection of edges and V(ζ) 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 ν, denoted as deg(ν), is the number of vertices connected to a fixed vertex ν. An edge e is represented by e=υωE(ζ).

2.1. Definitions

If ζ is a simple connected graph, the degree (deg(υ)) represents the count of different edges incident to any node within the closed neighborhood of υ. The vertex-edge degree (νε-degree) can be calculated by considering the number of distinct edges incident on any node υ within its open neighborhood. Moreover, the edge-vertex degree (εν-degree) of an edge eˇ is defined as the count of vertex unions between the open neighborhoods of the endpoints ω and υ. The εν-degree and νε-degree based Topological Indices (TIs) are presented below in mathematical notation.

The εν degree-based Zagreb index can be determined as follows: Mev(ζ)=eεEdegev(e)2. The 1st νεdegree Zagreb alpha index (M1αve(ζ)) is determined as: M1αve(ζ)=υεVdegev(υ)2. The 1st νεdegree Zagreb beta index (M1βve(ζ)) computed as: M1βve(ζ)=υωεEdegve(ω)+degve(υ). The second νεdegree Zagreb index (M2ve(ζ)) mathematically defined as: M2ve(ζ)=υωεE(degve(ω)×degve(υ)). The νεdegree Randić index (Rve(ζ)) mathematically satiated as: Rve(ζ)=υωεE(degve(ω)×degve(υ))12. The ενdegree Randić index (Rev(ζ)) determined as: Rev(ζ)=eεE(degve(e))12. The νεdegree Atom Bond Connectivity index (ABCve(ζ)) calculated by formula given below, as: ABCve(ζ)=υωεEdegve(ω)+degve(υ)2degve(ω)×degve(υ). The νεdegree Geometric Arithmetic index (GAve(ζ)) determined as: GAve(ζ)=ωυεE2degve(ω)×degve(υ)degve(ω)+degve(υ). The νεdegree Harmonic index (Hve(ζ)): Hve(ζ)=ωυεE(G)2degve(ω)+degve(υ). The νεdegree Sum-Connectivity index (Xve(ζ)) computed as: Xve(ζ)=ωυεE(G)(degve(ω)+degve(υ))12. Yamac and Cancan discuss this εν and νε 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 SiC4I[r,s].

4. 2D Structure of Silicon Carbide SiC4I[r,s]

The 2D molecular graph of SiC4I[r,s] is shown in Figure 1. 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 1 a unit cell and a structure of r=2 and s=1, r=3 and s=2 and r=s=3 are represented. Consequently, the total numbers of vertices, edges and faces in SiC4I[r,s] are; |V(SiC4I[r,s])|=10rs,|E(SiC4I[r,s])|=12rsrs,|F(SiC4I[r,s])|=2rsrs+2.

FIgure 1: 2-Dimensional Structure of SiC4-I[r,s], (A) Chemical Unit Cell of SiC4-I[r,s] (B) SiC4-I[3,3], (C) SiC4-I[2,1] (D) SiC4-I[3,2], Where Silicon Atoms Si are Blue and Carbon Atoms C Are Brown

5. Methodology of Silicon Carbide SiC4I[r,s] Formulas

The unit cell is used to compute silicon carbide formulae SiC4I[r,s]. To raise r, interconnect the unit cells horizontally, then connect the rows vertically to increase s. The connection points must be correct. Where r is the number of rows and s is the number of columns.

5.1. Vertex Partition

There are 3 kinds of nodes based on the degree of nodes. Vertices of 1st, 2nd and 3rd degree are represented as V1, V2 and V3 respectively as shown in Table 1.

Table 1: Vertex Partition of SiC4I[r,s]
[r,s] [1,1] [2,1] [3,1] [1,2] [2,2] [3,2] [1,3] [2,3] [3,3]
V1 3 6 9 3 6 9 3 6 9
V2 4 6 8 8 10 12 12 14 16
V3 3 8 13 9 24 39 15 40 65
Table 2: Degree of Vertex with Corresponding Cardinality
deg(ω) Cardinality
V1 3r
V2 2r+4s2
V3 10rs5r4s+2
Table 3: Vertex and Edges Frequency of SiC4I[r,s]
Total vertices Total edges
10rs 15rs4r2s+1

5.2. Edge Partition

By using above methodology we will partition the edges of SiC4I[r,s]. In the instance of SiC4I[r,s], there are five distinct edge portions, as shown in Table 4. It is important to note that the variables r,s1.

Table 4: ενDegree of SiC4I[r,s]
(deg(ω),deg(υ)) ενdegree Cardinality
(2,1) 3 2
(3,1) 4 3r2
(2,2) 4 r+2s2
(3,2) 5 2r+4s2
(3,3) 6 15rs10r8s+5

6. Main Results For Silicon Carbide SiC4I[r,s]

In this section, we calculate the main results for silicon carbide SiC4I[r,s]. 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: υεDegree of SiC4I[r,s] for all r,s1
deg(ω) υεdegree Cardinality
1 2 2
1 3 2r2
2 4 2
2 5 2r+4s4
3 6 2
3 7 3r
3 8 2r+4s6
3 9 10rs10r8s+8
Table 6: νεDegree of End Vertices of Each Edge of SiC4I[r,s]
deg(ω),deg(υ) ενdegree Cardinality
(3,1) (7,3) 3r2
(2,1) (4,2) 2
(2,2) (5,5) r+2s2
(2,2) (7,4) 1
(3,2) (7,5) 3
(3,2) (8,4) 1
(3,2) (8,5) 2r+4s7
(3,3) (8,7) r+1
(3,3) (8,8) s1
(3,3) (9,7) 5r3
(3,3) (9,8) 3r+6s11
(3,3) (9,9) 15rs19r15s+19
(i) The Mev index:

By making use of ενdegree of edge partitions of SiC4I[r,s], as shown in Table 5, we calculate the Mev index in the following lines: Mev(SiC4I[r,s])=eE(SiC4I[r,s])(degev(e)2)=2×32+(3r2)×42+(r+2s2)×42+(2r+4s2)×52+(15rs10r8s+5)×62=540rs246r156s+84.

(ii) The M1αve index:

By making use of νεdegree of vertices partition of SiC4I[r,s] for r,s2, as seen in Table 5, we compute the M1αve in the following lines: M1αve(SiC4I[r,s])=υV(SiC4I[r,s])(degve(υ)2)=2×22+(3r2)×32+2×42+(2r+4s4)×52+(3r)×72+(2r+4s6)×82+(10rs10r8s+8)×92=810rs458r292s+186.

(iii) The M1βve index:

By making use of νεdegree based partition of the end vertices of the edges of SiC4I[r,s] for r,s2, as shown in Table 6, we compute the M1βve in the following lines: M1βve(SiC4I[r,s])=ωυE(SiC4I[r,s])(degve(ω)+degve(υ))=(3r2)×10+6×2+(r+2s2)×10+1×11+3×12+(2r+4s7)×13+1×12+(5r3)×16+(3r+6s11)×17+(r+1)×15+(s1)×16+(15rs19r15s+19)×18=270rs130r80s+46.

(iv) The M2ve index:

Simply availing use of νεdegree based partition of end vertices of the edges of SiC4I[r,s] for r,s2, using Table 6, we compute the M2ve in the following lines: M2ve(SiC4I[r,s])=ωυE(SiC4I[r,s])(degve(ω)×degve(υ))=(3s2)×21+2×8+(r+2s2)×25+1×28+3×35+(2r+4s7)×40+1×32+(5r3)×63+(3r+6s11)×72+(r+1)×56+(s1)×64+(15rs19r15s+19)×81=1215rs784r509s+359.

(v) The Rve index:

By utilizing the νεdegree based partition of end vertices of the edges of SiC4I[r,s] for r,s2, as given in Table 6, we compute the Rve as follows, Rve(SiC4I[r,s])=ωυE(SiC4I[r,s])(degve(ω)×degve(υ))12=(3r2)×(21)12+2×(8)12+(r+2s2)×(25)12+1×(28)12+3×(35)12+(2r+4s7)×(40)12+1×(32)12+(5r3)×(63)12+(3r+6s11)×(72)12+(r+1)×(56)12+(s1)×(64)12+(15rs19r15s+19)×(81)12=53rs+(321+110+537+122+12148645)r+(210+12137120)s+(221127+12+3357210+142+131221162+1214+571360)=1.66rs+0.176r+0.197s+0.082.

(vi) The Rev index:

By utilizing the ενdegree of edges partition of SiC4I[r,s] for r,s2, as given in Table 5, we compute the Rev as follows, Rev(SiC4I[r,s])=eE(SiC4I[r,s])(degve(e)12)=2×(3)12+(3r2)×(4)12+(r+2s2)×(4)12+(2r+4s2)×(5)12+(15rs10r8s+5)×(6)12=156rs+(25106+2)r+(4586+1)s+(2325+562)=6.12rs1.18r0.47s+0.3015.

(vii) The ABCve index:

With the help of νεdegree based partition of the end vertices of the edges of SiC4I[r,s] for r,s2, as shown in Table 6, we compute the ABCve as follows, ABCve(SiC4I[r,s])=ωυE(SiC4I[r,s])degve(ω)+degve(υ)2degve(ω)×degve(υ)=(3r2)×821+2×848+(r+2s2)×825+1×928+3×1035+(2r+4s7)×1140+1032+(5r3)×1463+(3r+6s11)×1572+(r+1)×1356+(s1)×1464+(15rs19r15s+19)×1681=203rs+(6221+225+41110+523+3526+11327769)r+(425+81110+356+148203)s+(224221425+327+325141110+542+1526+11427148+769)=667rs+257r+6.06s4.911.

(viii) The GAve index:

By making use of νεdegree based partition of the end vertices of the edges of SiC4I[r,s] for r,s2, as shown in Table 6, we compute the GAve as follows, GAve(SiC4I[r,s])=ωυE(SiC4I[r,s])(2degve(ω)×degve(υ)degve(ω)+degve(υ))=(3r2)×22110+486+(r+2s2)×22510+2×2811+6×3512+(2r+4s7)×24013+23212+(5r3)×26316+(3r+6s11)×27217+(r+1)×25615+(s1)×26416+(15rs19r15s+19)×28118=15rs+(3215+81013+1578+36217+4141518)r+(161013+722172)s+(4232215+4711+4711+35228103+223+978132217+41413+16)=15rs4.35r+7.88s+8.213.

(ix) The Hve index:

By making use of νεdegree based partition of the end vertices of the edges of SiC4I[r,s] for r,s2, as shown in Table 6, we compute the Hve as follows, Hve(SiC4I[r,s])=ωυE(SiC4I[r,s])2degve(ω)+degve(υ)=2×(3r2)10+×46+2×(r+2s2)10+211+612+2×(2r+4s7)13+212+2×(5r3)16+2×(3r6s11)17+2×(r+1)15+2×(s1)16=53rs+858179560r+526326520s14721989=1.667rs+0.107r+0.198s+0.74.

(x) The Xve index:

By making use of νεdegree based partition of the end vertices of the edges of SiC4I[r,s] for r,s2, as shown in Table 6, we compute the Xve as follows, Xve(SiC4I[r,s])=ωυE(SiC4I[r,s])(degve(ω)+degve(υ))12=3r210+26+r+2s210+111+312+2r+4s713+112+3r316+3r+6s1117+r+115+s116+15rs19r15s+1918=52rs+(510+317+1151932+53)r+(610+61752+14)s+(261110+111+231117+115+19321)=3.53rs0.244r+0.067s0.137.

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 (r=0.7889), but the ev-degree Zagreb index shows excellent values of correlation as R=0.9808. Similarly, the correlation between properties of octane (acentric factor, boiling point, and entropy) and classical R-index is very low, as r(AF)=0.40484, r(S)=0.3506, and r(BP)=0.0737, but ev-degree-related Randić indices give amazing values of coefficients like R(AF)=0.8475, R(S)=0.8441, and R(BP)=0.7807. 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 SiC4I[r,s] based on ενdegree and νεdegree. The findings are extremely valuable and beneficial from both a chemical and pharmacological standpoint. In future, we can find ενdegree and νεdegree 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.

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