# Study of Vanadium Carbide Structures Based on Ve and Ev-Degree Topological Indices

Authors: Abdul Rauf; Saba Maqbool; Muhammad Naeem; Adnan Aslam; Hamideh Aram; Kraidi Anoh Yannick
Journal: Journal of Chemistry
 (2021)

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## Abstract

Vanadium is a biologically active product with significant industrial and biological applications. Vanadium is found in a variety of minerals and fossil fuels, the most common of which are sandstones, crude oil, and coal. Topological descriptors are numerical numbers assigned to the molecular structures and have the ability to predict certain of their physical/chemical properties. In this paper, we have studied topological descriptors of vanadium carbide structure based on ev and ve degrees. In particular, we have computed the closed forms of Zagreb, Randic, geometric-arithmetic, and atom-bond connectivity (ABC) indices of vanadium carbide structure based on ev and ve degrees. This kind of study may be useful for understanding the biological and chemical behavior of the structure.


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## Body

## 1. Introduction

Vertex degree concept has devised many topological indices that are applicable in QSPR/QSAR studies. Topological indices are widely used in theoretical and mathematical chemistry as they are associated with the topology of a chemical structure along with its other identical properties such as boiling points, strain energy, and stability [1]. In chemical graph theory, a chemical graph is referred as a molecular structure with atoms as its vertices and chemical bonds as its edges. A topological index is a numerical parameter that creates a link between the physical and chemical properties of a molecule [2]. Many topological descriptors based on degree have been introduced. These topological indices have provided assistance in calculating different parametric calculations related to molecular structures to make them understandable and beneficial. A lot of topological descriptors have been defined and studied so far, but Zagreb indices [3], Weiner index [4], and Randic index [5] are the most studied among all of them. To read more about the chemical applicability of topological descriptors, see [5–12].Researchers have attempted to study the varying behavior of transition metal carbides due to their complex structures. Such mineral metals are available in commercial places, and their salts are broadly utilized in our enterprises related to electrochemistry and material science. Among these, vanadium carbide complexes have shown crystal morphologies and stoichiometrics and display a great variety of superstructures. Very recently, many attempts have been done to purify high quality vanadium carbide by presenting different binary model system such asVC,V2C,V4C3,V6C5, and V8C7. For more details, see [13–16].LetG be a simple connected graph with its edge set and vertex set denoted by E and V, respectively. The neighbor set Nv of a vertex v contains those vertices v1 such that vv1∈E. The degree of vertex v is denoted by dv and is the cardinality of the set Nv. Let Nv=v1∈V:vv1∈E∪v be the closed neighborhood of v. To read more about the basic concepts related to graph, see [17].M. Chellali et al. [18] first introduced the concept of ev degree of an edge e and ve degree of a vertex v. The ev degree of an edge e is denoted by deve and is defined as the total number of vertices in the closed neighborhoods of the end vertices of an edge e. The ve degree of a vertex v is denoted by dvev and is the total number of edges that are adjacent with v and the first neighbor of v. Ediz [3] first introduced the concept of ve degree and ev degree Zagreb and Randic indices. The mathematical formulas of these indices are presented in Table 1. These newly defined indices were compared with Zagreb, Weiner, and Randic indices by modeling some of the physical/chemical properties of octane isomers. These indices have been observed to provide better correlation than the Randic, Weiner, and Zagreb indices for predicting some specific physical and chemical properties of octane isomers. Recently, a lot of work is done in the direction of computing newly defined ve degree and ev degree-based indices [19–23].Table 1
Mathematical formula of topological indices.
Topological indicesNotationMathematical formulaEv degree Randic indexRevG∑e∈EGdeve−1/2First Zagrebβ-indexM1βveG∑uv∈EGdveu+dvevEv degree Zagreb indexMevG∑e∈EGdeve2Second Zagrebβ-indexM2βveG∑uv∈EGdveu×dvevVe degree harmonic indexHveG∑uv∈EG2/dveu+dvevVe degree sum connectivity indexχveG∑uv∈EGdveu+dvev−1/2Ve degree geometric arithmetic indexGAveG∑uv∈EG2dveu×dvev/dveu+dvevVe degree atom bond connectivity indexABCveG∑uv∈EGdveu+dvev−2/dveu×dvev


## 2. Vanadium Carbide

Vanadium carbide belongs to the family of group IV to VI transition metal carbides and shows homogeneity to metal nitrides, monocarbides, and carbonitrides. They possess unique associations between physio-chemical properties such as high melting points, high temperature resistivity, strength, and hardness which are associated with good electrical and thermal conductivity. These rare combinations of properties make such compounds very interesting for the researchers. These materials can be used as wear-resistant hard alloys and as hard coatings for protection purposes, due to their nanochemical properties [24, 25].Vanadium carbide is the hardest inorganic metal-carbide with the formulaVC. VC is an incredibly hard refractory ceramic with exceptional wear resistance, high modulus of elasticity (400 GPa), and good strength retention even at high temperatures [26–28]. VC coatings are used in corrosion prevention, cutting tool application, machining, drilling, and dyeing. Some industrial uses of VC are given in [27, 29–31]. We denote the crystallographic structure of vanadium carbide by VCm,n. The molecular structure of vanadium carbide for m=5=n is depicted in Figure 1. The structure of VCm,n has total number of 3mn+m+n vertices and 6mn−m−n edges. Let Vi denote the vertex set containing the vertices of VCm,n of degree i. Then, the vertex set VVCm,n can be partitioned into six sets with V1=m+n+2, V2=2m+2n−4, V3=4m−2n+2, V4=1, V5=m+n−2, and V6=mn−m−n+1. Let Ei,j denote the edge set containing the edges of VCm,n with end vertices of degree i and degree j. The edge set of VCm,n can be partitioned based on the degree of end vertices as follows: E1,4 with 1 edge, E1,5 with 2 edges, E1,6 with m+n−1 edges, E2,4 with 2 edges, E2,5 with 2m+2n−4 edges, E2,6 with 2m+2n−6 edges, E3,4 with 1 edge, E3,5 with 3m+3n−8 edges, and E3,6 having 6mn−9m−9n+13 edges. In Theorem 1, we compute the ev degree Randic and ev degree Zagreb index of VCm,n.Figure 1
Molecular structure of vanadium carbide form=5 and n=5.


## 3. Main Results

Theorem 1.
Letm,n≥2, then(1)MevVCm,n=486mn−262m−262n+130,RevVCm,n=17mn+17+27+28+38−99m+17+27+28+38−99n+15+46−47−68−88+139.Proof.
To compute the ev degree Zagreb and ev degree Randic index ofVCm,n, we need to compute the ev degree of the edges in each partition set Ei,j. This calculation is presented in Table 2. Now, using the information presented in Table 2 and the definition of ev degree Zagreb and ev degree Randic index, we get(2)MevVCm,n=∑e∈EZdeve2=52E1,4+62E1,5+72E1,6+62E2,4+72E2,5+82E2,6+72E3,4+82E3,5+92E3,6=521+622+72m+n−l+622+722m+2n−4+822m+2n−6+921+723m+3n−8+826mn−9m−9n+13=486mn−262m−262n+130.RevVCm,n=∑e∈EZdeve−1/2,=5−1/2E1,4+6−1/2E1,5+7−1/2E1,6+6−1/2E2,4+7−1/2E2,5+8−1/2E2,6+7−1/2E3,4+8−1/2E3,5+9−1/2E3,6=5−1/21+6−1/22+7−1/2m+n−l+6−1/22+7−1/22m+2n−4+8−1/22m+2n−6+9−1/21+7−1/23m+3n−8+8−1/26mn−9m−9n+13=17mn+17+27+28+38−99m+17+27+28+38−99n+15+46−47−68−88+139.Table 2
Ev-degrees of edges of vanadium carbide.
du,dvdeveFrequencyE1,451E1,562E1,67m+n−1E2,462E2,572m+2n−4E2,682m+2n−6E3,47lE3,583m+3n−8E3,696mn−9m−9n+13Theorem 2.
Letm,n≥3, then(3a)M1βveVCm,n=216mn+76m+20n+54,M2βveVCm,n=1944mn−51m−1887n+1798,ABCveVCm,n=634324mn+1884+221130+224168+28121+227208+330252+32288+833306−1534324m+1884+221130+224168+28121+227208+330252+32288+233306−1534324n+1032+21455+31460−41884+21572+219110+220121−1021130+222117+223154+220120−102416822112−628121+225182−827208+225176+226187+229238−1430252−332288−1033306+26180+3734324,GAveVCm,n=1232436mn+28420+413023+416826+212130+420829+625232+228834+1630635−3032436m+28420+413023+416826+212130+420829+625232+228834+430635−3032436n+23212+45516+66016−88420+47217+411021+412122−2013023+411722+415425+412022−2016826+211222−1212130+418227−1620829+417627+418728+423831−2825232−628834−2030635+218028+7432436,HveVCm,n=13mn+1097490593929469607440m+39532069247643760n+801297510323950023374300,(3b)χveZ=mn+120+223+226+130+229+332+134+835−1536m+120+223+226+130+229+332+134+235−1536n+112+216+316−420+217+221+222−1023+222+225−1026+122−630+227−829+227+228+231−1432−334−1035+128+3736,RveVCm,n=6324mn+184+2130+2168+1221+2208+3252+1288+8306−15324m+184+2130+2168+1221+2208+3252+1288+2306−15324n+132+255+360−472+2110+2121−484−10130+2117+2154−10168+1112+6221+2182−8208+2176+2187+2238−14252−3288−10300+1180+37324.Proof.
To compute the ve degree-based indices, we need to find the edge partition ofEVCm,n based on the ve degree of end vertices of each edge. This partition is presented in Table 3. Now, using the values from Table 3 and the definition of ve degree indices, we get(4)M1βveVCm,n=∑uv∈EVCm,ndveu+dvev=12E1+16E2+16E3+20E4+17E5+21E6+22E7+23E8+22E9+25E10+22E11+26E12+22E13+30E14+27E15+29E16+27E17+28E18+31E19+32E20+34E21+35E22+28E23+36E24=121+162+163+20m+n−4+172+212+222+232m+2n−10+222+252+222+262m+2n−10+221+30m+n−6+272+292m+2n−8+272+282+312+323m+3n−14+34m+n−3+358m+2n−10+281+366mn−15m−15n+37=216mn+76m+20n+54,M2βveVCm,n=∑uv∈EVCm,ndveu×dvev=32E1+55E2+60E3+84E4+72E5+110E6+168E12+112E13+121E7+130E8+117E9+154E10+120E11+121E14+182E15+208E16+176E17+187E18+238E19+252E20+288E21+306E22+180E23+324E24=321+552+603+84m+n−4+722+1102+1212+1302m+2n−10+1172+1542+1202+1682m+2n−10+1121+121m+n−6+1822+2082m+2n−8+1762+1872+2382+2523m+3n−14+288m+n−3+3068m+2n−10+1801+3246mn−15m−15n+37=1944mn−51m−1887n+1798.(5a)ABCveVCm,n=∑uv∈EVCm,ndveu+dvev−2dveu×dvev=1032E1+1455E2+1460E3+1884E4+21130E8+1572E5+19110E6+20121E7+22117E9+23154E10+20120E11+24168E12+22112E13+28121E14+25182E15+27208E16+25176E17+26187E18+29238E19+30252E20+32288E21+33306E22+26180E23+34324E24=634324mn+1884+221130+224168+28121+227208+330252+32288+833306−1534324m+1884+221130+224168+28121+227208+330252+32288+233306−1534324n+1032+21455+31460−41884+21572+219110+220121−1021130+222117+223154+220120−1024168+22112−628121+225182−827208+225176+226187+229238−1430252−332288−1033306+26180+3734324,(5b)GAveVCm,n=∑uv∈EVCm,n2dveu×dvevdveu+dvev=23212E1+25516E2+26016E3+28420E4+27217E5+211021E6+212122E7+213023E8+211722E9+215425E10+212022E11+216826E12+211222E13+212130E14+218227E15+220829E16+217627E17+218728E18+223831E19+225232E20+228834E21+230635E22+218028E23+232436E24=1232436mn+28420+413023+416826+212130+420829+625232+228834+1630635−3032436m+28420+413023+416826+212130+420829+625232+228834+430635−3032436n+23212+45516+66016−88420+47217+411021+412122−2013023+411722+415425+412022−2016826+211222−1212130+418227−1620829+417627+418728+423831−2825232−628834−2030635+218028+7432436.(6a)HveVCm,n=∑uv∈EVCm,n2dveu+dvev=212E1+216E2+216E3+220E4+217E5+221E6+222E7+223E8+222E9+225E10+222E11+226E12+222E13+230E14+227E15+229E16+227E17+228E18+231E19+232E20+234E21+235E22+228E23+236E24=13mn+1097490593929469607440m+39532069247643760n+801297510323950023374300,(6b)χveVCm,n=∑uv∈EVCm,ndveu+dvev−1/2=12−1/2E1+16−1/2E2+16−1/2E3+20−1/2E4+17−1/2E5+21−1/2E6+22−1/2E7+23−1/2E8+22−1/2E9+25−1/2E10+22−1/2E11+26−1/2E12+22−1/2E13+30−1/2E14+27−1/2E15+29−1/2E16+27−1/2E17+28−1/2E18+31−1/2E19+32−1/2E20+34−1/2E21+35−1/2E22+28−1/2E23+36−1/2E24=mn+120+223+226+130+229+332+134+835−1536m+120+223+226+130+229+332+134+235−1536n+112+216+316−420+217+221+222−1023+222+225−1026+122−630+227−829+227+228+231−1432−334−1035+128+3736.(7)RveVCm,n=∑uv∈EVCm,ndveu×dvev−1/2=32−1/2E1+55−1/2E2+60−1/2E3+84−1/2E4+72−1/2E5+110−1/2E6+121−1/2E7+130−1/2E8+117−1/2E9+154−1/2E10+120−1/2E11+168−1/2E12+112−1/2E13+121−1/2E14+182−1/2E15+208−1/2E16+176−1/2E17+187−1/2E18+238−1/2E19+252−1/2E20+288−1/2E21+306−1/2E22+180−1/2E23+324−1/2E24=6324mn+184+2130+2168+1221+2208+3252+1288+8306−15324m+184+2130+2168+1221+2208+3252+1288+2306−15324n+132+255+360−472+2110+2121−484−10130+2117+2154−10168+1112+6221+2182−8208+2176+2187+2238−14252−3288−10300+1180+37324.Table 3
Edge Partition of vanadium carbide.
Edgedveu,dvevFrequencyE1(4, 8)1E2(5, 11)2E3(6, 10)3E4(6, 14)m+n−4E5(9, 8)2E6(10, 11)2E7(11, 11)2E8(10, 13)2m+2n−10E9(9, 13)2E10(11, 14)2E11(12, 10)2E12(12, 14)2m+2n−10E13(8, 14)1E14(17, 13)m+n−6E15(14, 13)2E16(16, 13)2m+2n−8E17(16, 11)2E18(17, 11)2E19(17, 14)2E20(18, 14)3m+3n−14E21(16, 18)m+n−3E22(17, 18)2m+2n−10E23(18, 10)1E24(18, 18)6mn−15m−15n+37


## 4. Numerical Results and Discussion

Topological indices are used as vital tools for the analysis of chemicals, given the essential topology of chemical structures. Zagreb-type indices are used to calculate the totalπ−electronic energy of molecules [32]. The Randich index is commonly used to determine the chemical similarity of molecular compounds, as well as to calculate the boiling point and Kovaz constants of molecules [33]. The atom-bond connectivity index (ABC) provides a very good correlation for calculating strain energies as well as for the stability of linear and branched chemical structures [34]. It can be seen from Table 4 and 5 and Figures 2–6 of indices, an increase in the value of m and n raises the values of topological descriptors for vanadium carbide structure.Table 4
Numerical results of indices for vanadium carbide.
m,nMevHRevHM1αveHM1βveHM2βveH1,1920.133 21324211018042,210261.070 42732285456983,329322.763 64788403013 4804,458105.212 67492563825 1505,596608.417 710 844767840 7086,614 48212.378 614 84410 15060 1547,720 27617.095 419 49213 05483 4888,827 04222.568 224 78816 390110 7109,934 78028.796 930 73220 158141 82010,1043 49035.781 637 32424 358176 818Table 5
Numerical results of indices for vanadium carbide.
m,nABCveHGAveHHveHχveHRveH1,14.707 410.857 01.708 81.855 01.031 92,213.289 632.034 63.240 96.122 72.722 33,325.759 065.212 25.439 612.390 35.079 34,442.115 7110.389 88.305 020.657 98.103 05,562.359 7167.567 411.837 030.925 511.793 36,686.491 0236.745 016.035 743.193 216.150 37,7114.509 6317.922 620.901 157.460 821.174 08,8146.415 5411.100 226.433 273.728 426.864 39,9182.208 7516.277 932.631 991.996 133.221 310,10221.889 3633.455 539.497 3112.263 740.245 0Figure 2
3D plot of (a)MevVCm,n (b) RevVCm,n.
(a)(b)Figure 3
3D plot of (a)M1βveVCm,n (b) M2βveVCm,n.
(a)(b)Figure 4
3D plot of (a)ABCveVCm,n (b) GAveVCm,n.
(a)(b)Figure 5
3D plot of (a)HveVCm,n (b) χveVCm,n.
(a)(b)Figure 6
Graphical representation ofRveVCm,n.


## 5. Conclusion

Graphs invariants are calculated by some well-known topological indices which are important tools for resembling and forecasting the properties of chemical compounds in QSPRs and the QSARs. The TI is a numerical measure that represents the biological, physical, and chemical properties of molecules such as boiling, melting, and flickering point; moisture; and forming heat. In this paper, we have computed the ev degree and ve degree-based topological indices with graphical representations for the molecular structure of vanadium carbide for a better understanding of pharmaceutical, physical, chemical, and biological properties.



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