# Topological Descriptors of M-CarbonMr,s,t

Authors: Hong Yang; Muhammad Naeem
Journal: Journal of Chemistry
 (2021)

---

## Abstract

We have studied topological indices of the one the hardest crystal structures in a given chemical system, namely, M-carbon. These structures are based and obtained by the famous algorithm USPEX. The computations and applications of topological indices in the study of chemical structures is growing exponentially. Our aim in this article is to compare and compute some well-known topological indices based on degree and sum of degrees, namely, general Randić indices, Zagreb indices, atom bond connectivity index, geometric arithmetic index, new Zagreb indices, fourth atom bond connectivity index, fifth geometric arithmetic index, and Sanskruti index of the M-carbonMr,s,t. Moreover, we have also computed closed formulas for these indices.


---

## Body

## 1. Introduction

One of the hardest structures of carbon is diamond. In 2011, Andriy and Artem searches 9500 structures of different system sizes, and they produced a large number of superhard allotropes; these allotropes being as hard as diamonds [1]. One of the superhard carbon allotropes that they studied was M-carbon (Figure 1). Scientists also believe that the synthesis and practical applications of some of these structures may be possible. Some studies also exist giving indications that these types of carbon allotropes such as M-carbon have been obtained by applying cold compression on graphite [2, 3].Figure 1
The M-carbon structureM[3,3,2].In this study, we intend to study and compute the degree-based topological indices of M-carbon structures. One of the first and very old topological indices is the Wiener index [4], and this index is also known as the path number. After that, the scientists of various field started exploring this new technique to study chemical and physical properties of chemical structure, compounds, and molecules. A list of topological indices that we shall discuss in this study is given in Table 1, which includes Randić index, general Randić indices, Zagreb indices, atom bond connectivity index, geometric arithmetic indices, new Zagreb indices, fourth atom bond connectivity index, fifth geometric arithmetic index, and Sanskruti index. For some literature study and results related to these indices, see [5–13].Table 1
Famous degree-based topological indices.
S. no.Topological indexNotationFormulaAuthors1Randić indexR−1/2G∑oℓ∈EG1/dodℓRandić [14] in 19752General Randić indexRαG∑oℓ∈EGdodℓαBollobás and Erdös [15] and Amic et al. [16] in 19883Atom bond connectivity indexABCG∑oℓ∈EGdo+dℓ−2/dodℓEstrada et al. [17]4Geometric arithmetic indexGAG∑oℓ∈EG2dodℓ/do+dℓVukićevič and Furtula. [18]5First ZagrebM1G∑oℓ∈EGdo+dℓGutman and Trinajstić in 1972[19]6Second ZagrebM2G∑oℓ∈EGdo×dℓGutman and Das in 2004 [20]7New degree-based Zagreb indexHMG∑oℓ∈EGdo+dℓ2Shirdel et al. [21] in 20138First Zagreb coindexM1¯G∑oℓ∉EGdo+dℓIn 2008, Došlić [22]9Second Zagreb coindexM2¯G∑oℓ∉EGdodℓSame as above10Fourth atom bond connectivity indexABC4G∑oℓ∈EGSo+Sℓ−2/SoSℓGhorbhani and Hosseinzadeh [23]11Fifth version of geometric arithmetic indexGA5∑oℓ∈EG2SoSℓ/So+SℓGraovoc et al. [24]12Sanskruti indexSG∑oℓ∈EGSo×Sℓ/So+Sℓ−23In 2016, Hosamani [25]In Table1, the number So represents the sum ∑oℓ∈EGdℓ and the number Sℓ represents the sum ∑ℓo∈EGdo, where do is the degree of vertex o and dℓ represents the degree of vertex ℓ. In 2016, Gutman et al. [26] proved the following theorems for some of the indices in Table 1.Theorem 1 (See [26]).
LetG be a graph with VG vertices and EG edges. Then,(1)M1¯G=2EG|VG−1−M1G.Theorem 2 (See [26]).
LetG be a graph with VG vertices and EG edges. Then,(2)M2¯G=2EG2−12M1G−M2G.


## 2. Construction ofMr,s,t for Topological Study

In this section, we shall present our main results about the M-carbon structure denoted asMr,s,t. First, we need to give a brief explanation of the variables r,s,t in the notation Mr,s,t. To find and compute the topological indices of the M-carbon structure, we have introduced a way of constructing its structure by the means of these three variables, where r represents the unit as shown in Figures 2(a) and 2(b) represents a chain containing three units, where the connection (bond) is shown in blue color. The variable s represents the number of connected chains with each having r numbers of units (Figure 3). The variable t represents the number of connected layers. There are two types of layers odd layer for t=1,3,5,… and even layers for t=2,4,6,…, both are generated by different unit cells. The one depicted in Figure 3 is the odd layer (that is for t=1,3,5,…) which was generated by the unit of Figure t=2,4,6,…). The unit cell of an even layer is shown in Figure 4(a), the chain in even layer is shown in Figure 4(b), and Figure 4(c) depicts an even layer. Then, finally, the M-carbon structure Mr,s,t is shown in Figure 5, which also depicts how two layers, an even and odd, are connecting. In Figure 5, these connections (bonds) between two layers are shown in red colour. So, in this way, we get structure of M-carbon (Figure 1). By our construction, the graph of M-carbon Mr,s,t,r≥2,s≥2,t≥2 consists of 8rst number vertices and 16rst−4rt−5st−2rs+t+s number of edges.Figure 2
(a) The M-carbon unit cell. (b) Chain of three units and units connected by blue edges.
(a)(b)Figure 3
The odd layer (fort=1) of three chains and bonding of chains is shown by blue edges.Figure 4
Even layer construction. (a) The unit cell. (b) Chain of three units. (c) The layer of three chains.
(a)(b)(c)Figure 5
The structureM3,3,2. The connection of two layers. Bonding is shown is red color.


## 3. Main Results

The graphMr,s,t has 2s+1t,4r−2+2s−1t,2rst+2rs+6rt−st−6r−s−3t+3, and 6rst−2rs−6rt−3st+2r+s+3t−1 vertices of degrees 1,2,3, and 4, respectively. The degree-based edge partition of Mr,s,t is given in Table 2.Table 2
Degree-based edge partition ofMr,s,t for r,s,t≥2.
du,dvFrequency(1, 2)2t + 2(1, 3)2st−2(2, 2)2(2, 3)8r + 2s + 2t−8(2, 4)4st−4s−6t + 6(3, 3)6rt + 2sr−4r−2s−3t + 5(3, 4)4rs + 4rt−2s−4t + 2(4, 4)16rst−8rs−14rt−11st−4r + 7s + 10t−7In all the theorems in the following, we used Maple for the computations of mathematical expression and graphical comparisons.Theorem 3.
LetG be the chemical structural graph of M-carbon Mr,s,t, and the general Randić index of Mr,s,t is given by(3)RαMr,s,t=2×4a−7×16a+2×2at+8×6ar+2×6as+2×6at−4×8as−6×8at−4×9ar−2×9as−3×9at−2×12as−4×12at−4×16ar+7×16as+10×16at−8×6a−2×3a+2×2a+2×3ast+4×8ast+2×9ars+6×9art+4×12ars+4×12art−8×16ars−14×16art−11×16ast+5×9a+16×16arst+6×8a+2×12a.Proof.
LetG be the chemical structural graph of M-carbon Mr,s,t; then, by using serial number 2 of Table 1 and edge partition given in Table 2, the general Randić index RαG of Mr,s,t is computed as follows:(4)RαMr,s,t=∑oℓ∈EGdo×dℓα=2t+21×2α+2st−21×3α+22×2α+8r+2s+2t−82×3α+4st−4s−6t+62×4α+6rt+2rs−4r−2s−3t+53×3α+4rs+4rt−2s−4t+23×4α+16rst−8rs−14rt−11st−4r+7s+10t−74×4α.
The result follows after some simple computations from this equation.
It is very simple and clear to see that from the above Theorem3, the following Corollary is true.Corollary 1.
(5)RαMr,s,t=256rst−62rs−122rt−138st−52r+50s+53t−37ifα=123r36−17s144+23st48+13t24+rt8+rs18+rst+77144ifα=−123st+48st+29rs+1616rst−23ifα=12+22+22t+86r+26s+26t−48s−68t−49r−29s−39t−212s−412t−416r+716s+1016t+24−86+68+59+212−716+69rt+412rs+412rt−1116st−1416rt−816rs23st3+8st2+29rs9+16rst−233+2ifα=−12+2t+46r3+6s3+6t3−8s2−38t4−49r9−29s9−9t3−12s6−12t3−16r4+716s16+516t8+42−463+384+599+126−71616+29rt3+12rs3+12rt3−1116st16−716rt8−16rs2.Corollary 2.
The second Zagreb indexM2Mr,s,t is the same as RαMr,s,t for α=1, so M2Mr,s,t can be obtained from Corollary 1.Theorem 4.
The first Zagreb indexM1Mr,s,t of M-carbon structure is given by(6)M1Mr,s,t=128rst−24rs−48rt−56st−16r+16s+14t−10.Proof.
LetG be the chemical structural graph of M-carbon Mr,s,t; then, by using serial number 5 of Table 1 and edge partition given in Table 2, the first Zagreb index RαG of Mr,s,t is computed as follows:(7)M1Mr,s,t=∑oℓ∈EGdo+dℓ=2t+21+2+2st−21+3+22+2+8r+2s+2t−82+3+4st−4s−6t+62+4+6rt+2rs−4r−2s−3t+53+3+4rs+4rt−2s−4t+23+4+16rst−8rs−14rt−11st−4r+7s+10t−74+4.
Thus, the result follows by simple calculations.
The next theorem gives us the new degree-based Zagreb index as defined in [21].Theorem 5.
The new degree-based Zagreb indexHMMr,s,t of M-carbon structure is given by(8)HMMr,s,t=1024rst−244rs−484rt−528st−200r+184s+188t−136.Proof.
LetG be the chemical structural graph of M-carbon Mr,s,t; then, by using serial number 7 of Table 1 and edge partition given in Table 2, the new degree-based Zagreb index HM of Mr,s,t is computed as follows:(9)HMMr,s,t=∑oℓ∈EGdo+dℓ2=2t+21+22+2st−21+32+22+22+8r+2s+2t−82+32+4st−4s−6t+62+42+6rt+2rs−4r−2s−3t+53+32+4rs+4rt−2s−4t+23+42+16rst−8rs−14rt−11st−4r+7s+10t−74+42.
Thus, the result follows by simple calculations.
In the next two theorems, we shall compute the newly defined Zagreb coindex indices which are defined in the form of nonedges of a chemical graph.Theorem 6.
The first Zagreb coindex indexM1¯Mr,s,t of M-carbon structure is given by(10)M1¯Mr,s,t=256r2s2t2−32r2s2t−64r2st2−80s2t2r+16s2rt+16t2rs−160rst+28rs+56rt+66st+16r−18s−16t+10.Proof.
The first Zagreb coindex index of chemical structural graph of M-carbonMr,s,t can be computed by using both serial number 8 of Table 1 and Theorem 1. It is explained and calculated as follows:(11)M1¯Mr,s,t=2EMr,s,t|VMr,s,t−1−M1Mr,s,t=216rst−4rt−5st−2rs+t+s8rst−1−128rst−24rs−48rt−56st−16r+16s+14t−10.
Thus, the result follows by simple calculations.Theorem 7.
The second Zagreb coindex indexM2¯Mr,s,t of M-carbon structure is given by(12)M2¯Mr,s,t=512r2s2t2−128r2s2t−256r2st2−320s2t2r+8r2s2+32r2st+32r2t2+104s2rt+144t2rs+50s2t2−8s2r−344rst−16rt2−20s2t−20st2+74rs+146rt+2s2+170st+2t2+60r−58s−60t+42.Proof.
The second Zagreb coindex index of chemical structural graph of M-carbonMr,s,t can be computed by using both serial number 9 of Table 1 and Theorem 2. It is explained and calculated as follows:(13)M2¯Mr,s,t=2EMr,s,t2−12M1Mr,s,t−M2Mr,s,t=216rst−4rt−5st−2rs+t+s2−1/2128rst−24rs−48rt−56st−16r+16s+14t−10−256rst−62rs−122rt−138st−52r+50s+53t−37.
Thus, the result follows by simple calculations.
In the coming two theorems, we shall find closed formulas for the ABC and GA indices ofMr,s,t M-carbon structure.Theorem 8.
Consider the graphG≅Mr,s,t of M-carbon with r≥2,s≥2,t≥2; then, its ABC index is equal to(14)ABCMr,s,t=+2rs153+2rt153−2rs6−7rt62−25st612+4rt+4rs3−s153−2t153+4rst6−29612−2t−2s+103+153+22st−6r+76s4+56t2+2+42r−8r3−4s3−2t.Proof.
LetG be the chemical structural graph of M-carbon Mr,s,t; then, by using serial number 3 of Table 1 and edge partition given in Table 2, the ABC index of Mr,s,t is computed as follows:(15)ABCMr,s,t=∑oℓ∈EGdo+dℓ−2dodℓ=2t+21+2−21×2+2st−21+3−21×3+22+2−22×2+8r+2s+2t−8+2+3−22×3+4st−4s−6t+62+4−22×4+6rt+2rs−4r−2s−3t+53+3−23×3+4rs+4rt−2s−4t+23+4−23×4+16rst−8rs−14rt−11st−4r+7s+10t−74+4−24×4.
The results now follow after some simple computations of above expression.Theorem 9.
Consider the graphG≅Mr,s,t of M-carbon with r≥2,s≥2,t≥2; then, its GA index is equal to(16)GAMr,s,t=16rst−11st−82t3+1623+3st+37+166r5+46s5+46t5−1665+82st3−82s3−6rs−8rt−8r+5s+7t+163rs7+163rt7−83s7−163t7.Proof.
LetG be the chemical structural graph of M-carbon Mr,s,t; then, by using serial number 4 of Table 1 and edge partition given in Table 2, the GA index of Mr,s,t is computed as follows:(17)GAMr,s,t=∑oℓ∈EG2do×dℓdo+dℓ,2t+221×21+2+2st−221×31+3+222×22+2+8r+2s+2t−822×32+3+4st−4s−6t+622×42+4+6rt+2rs−4r−2s−3t+523×33+3+4rs+4rt−2s−4t+223×43+4+16rst−8rs−14rt−11st−4r+7s+10t−724×44+4.
The results now follow after some simple computations of the above expression.
Table3 provides the edge partition of the M-carbon structure Mr,s,t on the bases of sum of the degrees in the open neighbourhood of the end vertices of each an edge for each edge of Mr,s,t. The first of such type of indices were introduced by Ghorbhani and Hosseinzadeh [23], and then, another one was introduced by Hosamani [25]. These are defined in Table 1.
In the following three theorems, we gave closed formula for the three indices, namely, fourth atom bond connectivity indexABC4G, fifth version of geometric arithmetic index GA5G, and Sanskruti index SG for the graph of G≅Mr,s,t.Table 3
Sum degree-based edge partition ofMr,s,t for r,s,t≥3.
So,SℓFrequencySo,SℓFrequency(3, 7)4(3, 8)2(4, 11)2t(4, 12)2st−2s−4t + 6(4, 13)2st−2s−4t + 4(5, 5)2(5, 7)2(5, 8)2(6, 7)2(6, 8)4s−10(6, 9)8r−8(7, 8)2t−2(7, 11)2(7, 12)2t−2(7, 15)2(8, 8)2t−2(8, 10)2t−2(8, 11)2t−2(8, 12)4st−10t + 6(8, 14)2(8, 15)2t−4(9, 10)8r−8(9, 12)2(9, 14)2r−2(9, 15)4r−6(10, 10)4rs + 6rt−16r−4s−8t + 18(10, 11)2(10, 14)4r−6(10, 15)4rs + 4rt−12r−4s−4t + 12(11, 12)2t−2(11, 15)2t−2(11, 16)2(12, 13)2st−2s−4t + 4(12, 15)2t(12, 16)2st + 2s−4t−6(13, 14)2t−2(13, 15)4st−4s−10t + 10(14, 15)4r−4(14, 16)4r−6(15, 15)6rt−2r−7t + 1(15, 16)10rs−28r−10s + 4rt−4t + 30(16, 16)16rst−24rt−20rs−21st + 38t + 23s + 24r−40Theorem 10.
TheABC4 index of the graph G≅Mr,s,t of M-carbon with r≥3,s≥3,t≥3 is given by(18)ABC4G=3+3st+46221+11t515+t14311−377035+209055−255+332+47210−62s5−53t2+5718−35721+51122+1428−122t5+6r3+t144+t7711+4r33045−47r15−147t15+2r77035+6r7035+t7010−s426−t423+23t182182+3r302+281s30240+41t3024+4t11055+4r17015−2r1385−2s13815−2t13815+4r789+s7812−t786+t37422+t35721−7r43515−s4356−t43515−s19513−2t19513−s89739−2t89739−5rs304−3rt302−251st30240+2rs13815+2rt13815+st7812+66+rst30+374242−37422+rs4356+rt43515+st19513+st89739+st426+62rs5+92rt5+47rt5−417015+42121+219513+21385+7777+3110110−11306+357+4352+2715−137035−23182182−257836+289739−192r5−233015.Proof.
LetG be the chemical structural graph of M-carbon Mr,s,t; then, by using serial number 10 of Table 1 and edge partition given in Table 3, the ABC index of Mr,s,t is computed as follows:(19)ABC4Mr,s,t=∑oℓ∈EGSo+Sℓ−2SoSℓ=43+7−23×7+23+8−23×8+2t4+11−24×11+2st−2s−4t+64+12−24×12+2st−2s−4t+44+13−24×13+25+5−25×5+25+7−25×7+25+8−25×8+26+7−26×7+26+8−26×8+8r−86+9−26×9+2t−27+8−27×8+27+11−27×11+2t−27+12−27×12+27+15−27×15+2t−28+8−28×8+2t−28+10−28×10+2t−28+11−28×11+4st−10t+68+12−28×12+28+14−28×14+2t−48+15−28×15+8r−89+10−29×10+29+12−29×12+2r−29+14−29×14+4r−69+15−29×15+4rs+6rt−16r−4s−8t+1810+10−210×10+210+11−210×11+4r−610+14−210×14+4rs+4rt−12r−4s−4t+1210+15−210×15+2t−211+12−211×12+2t−211+15−211×15+211+16−211×16+2st−2s−4t+412+13−212×13+2t12+15−212×15+2st+2s−4t−612+16−212×16+2t−213+14−213×14+4st−4s−10t+10×13+15−213×15+4r−414+15−214×15+4r−614+16−214×16+6rt−2r−7t+115+15−215×15+10rs−28r−10s+4rt−4t+3015+16−215×16+16rst−24rt−20rs−21st+38t+23s+24r−4016+16−216×16.
The results now follow after some simple computation of the above expression.Theorem 11.
Consider the graphG≅Mr,s,t of M-carbon with r,s,t≥3; then, its GA5 index is given by(20)GA5G=−21+6r+19s+153st7+16t59−21st−859+1337+25t+4195+161317+3s7−303t7+161127−91614345+548r14345+2r353+8t2119+48r1019+8t2219−80s1531−32t1531−86r5−86s5−286t5−12rt−16rs+8t1415−821029−216513+4t18227+8t3023−2s1957−5t1957+52611+8st3925+813st17+44213+2st1957+16rst+210511+8t1115+2t16513−82119+51957−813s17−1613t17+3871562−82219−47210247+2779+411021−163023−2353+8711−418227+8r21029+163925−83323+8t3323−8s3925−16t3925−193r1531+80rs1531+32rt1531+8rs65+8rt65+8st65.Proof.
LetG be the chemical structural graph of M-carbon Mr,s,t; then, by using serial number 11 of Table 1 and edge partition given in Table 3, the GA index of Mr,s,t is computed as follows:(21)GA5Mr,s,t=∑oℓ∈EG2So×SℓSo+Sℓ=423×73+7+223×83+8+2t24×114+11+2st−2s−4t+624×124+12+2st−2s−4t+424×134+13+225×55+5+225×75+7+225×85+8+226×76+7+226×86+8+8r−826×96+9+2t−227×87+8+227×117+11+2t−227×127+12+227×157+15+2t−228×88+8+2t−228×108+10+2t−228×118+11+4st−10t+628×128+12+228×148+14+2t−428×158+15+8r−829×109+10+229×129+12+2r−229×149+14+4r−629×159+15+4rs+6rt−16r−4s−8t+18210×1010+10+2210×1110+11+4r−6210×1410+14+4rs+4rt−12r−4s−4t+12210×1510+15+2t−2211×1211+12+2t−2211×1511+15+2211×1611+16+2st−2s−4t+4212×1312+13+2t212×1512+15+2st+2s−4t−6212×1612+16+2t−2213×1413+14+4st−4s−10t+10×213×1513+15+4r−4214×1514+15+4r−6214×1614+16+6rt−2r−7t+1215×1515+15+10rs−28r−10s+4rt−4t+30215×1615+16+16rst−24rt−20rs−21st+38t+23s+24r−40216×1616+16.
The results now follow after some simple computation of the above expression.Theorem 12.
Consider the graphG≅Mr,s,t of M-carbon with r,s,t≥3; then, its Sanskruti index SG is given by(22)SG=c1r+c2s−c0−c4st+c2t−c5rt−c6rs+c7rst,where c0=1774341472659763130123583358813576506563/233972495845970544399543494688000000, c1=3841015904100770126879995213/387633527028142427996964000, c2=42042269311169825975177/8150780755142420850,c3=33779105325980576848378857231211/3203582868001177090884000000,c4=566977749527701739/61888713909750,c5=731518633569104009441/98932250100348000,c6=26845875203036608/5408104050675 and,c7=33554432/3375.Proof.
LetG be the chemical structural graph of M-carbon Mr,s,t; then, by using serial number 12 of Table 1 and edge partition given in Table 3, the Sanskruti index SMr,s,t of Mr,s,t is computed as follows:(23)GA5Mr,s,t=∑oℓ∈EGSo×SℓSo+Sℓ−23=43×73+7−23+23×83+8−23+2t4×114+11−23+2st−2s−4t+64×124+12−23+2st−2s−4t+44×134+13−23+25×55+5−23+25×75+7−23+25×85+8−23+26×76+7−23+26×86+8−23+8r−86×96+9−23+2t−27×87+8−23+27×117+11−23+2t−27×127+12−23+2×157+15−23+2t−28×88+8−23+2t−28×108+10−23+2t−28×118+11−23+4st−10t+68×128+12−23+28×148+14−23+2t−48×158+15−23+8r−89×109+10−23+29×129+12−23+2r−29×149+14−23+4r−69×159+15−23+4rs+6rt−16r−4s−8t+1810×1010+10−23+210×1110+11−23+4r−610×1410+14−23+4rs+4rt−12r−4s−4t+1210×1510+15−23+2t−211×1211+12−23+2t−211×1511+15−23+211×1611+16−23+2st−2s−4t+412×1312+13−23+2t12×1512+15−23+2st+2s−4t−612×1612+16−23+2t−213×1413+14−23+4st−4s−10t+10×13×1513+15−23+4r−414×1514+15−23+4r−614×1614+16−23+6rt−2r−7t+115×1515+15−23+10rs−28r−10s+4rt−4t+3015×1615+16−23+16rst−24rt−20rs−21st+38t+23s+24r−4016×1616+163.
The results now follow after some simple calculations.


## 4. Comparison and Discussion

In this section, we shall discuss and compare all the computed indices (degree based and sum degree based) from Theorems3–12 both graphically and numerically (by a tables).Tables4 and 5 provide us a numerical comparison. From these two tables and computations, we can analyse that which type of indices are closed related and can be used to study the same type of chemical, biochemical, physical, and biophysical properties of the structure of Mr,s,t. We can see that the values of GA and GA5 are very close and almost similar, so we can purpose that any one of them can be used in place of each other for the study of chemical structure Mr,s,t. They also show the same kind of behaviour in graphical analysis (Figure 6). On the same pattern, we can also see the closely related values of R−1, R−1/2, ABC4, R1/2, ABC, GA, and GA5, and these values are also produced in an increasing order, respectively. So, we can purpose to use those indices which give smaller values for the study of respective properties of structure Mr,s,t giving smaller values after the experimental study and the graphical comparison in Figure 7 which also shows the same type of patterns. Similarly, the indices M1, R1,HM,S,M1¯, and M2¯ give us larger values even for such small initial inputs of r,s,t. Such behaviour shows that (and we purpose that) they cannot be used as closely related to the previous mentioned indices to study the same type of properties of Mr,s,t. We can also see the behaviour of these indices in Figures 8 and 9. It is purposed to use these indices in the study of such properties of Mr,s,t which gives higher experimental values.Table 4
Numerical comparison of indices ofMr,s,t for some initial values of r,s,t≥2.
r,s,tR1R−1R1/2R−1/2M1M1¯M2¯HM(2, 2, 2)82513.3260.932.453010558143983376(3, 3, 3)413036.71157.6106.22336143 434224554416724(4, 4, 4)11399791157.62506190868642145097845992(5, 5, 5)24168147.36397.8489.6128603453670598985297324(6, 6, 6)43973246.611510847231141058757418818838176864(7, 7, 7)723503831879613391347377202.72×107290756(8, 8, 8)110835563286362013574466.14×1071.1×108445144(9, 9, 9)160964792414182871830601.2×1082.3×108646172(10, 10, 10)22427310775752539431153302.3×1084.4×108899984Table 5
Numerical comparison of indices ofMr,s,t for some initial values of r,s,t≥2.
r,s,tABCGAABC4GA5S(2, 2, 2)59.386.2———(3, 3, 3)220334.6138.8334.7114310(4, 4, 4)546.8847.8330.6843.3347250(5, 5, 5)10981721651.81712.2775750(6, 6, 6)19333052113530371.4595×106(7, 7, 7)31104936181349142.46×106(8, 8, 8)46887469272074403.8×106(9, 9, 9)6725107473887107115.6×106(10, 10, 10)9282148655348148207.9×106Figure 6
Comparison ofGA index and GA5 index of Mr,s,t for fix t=1. Red color represents GA, and green color represents GA5.Figure 7
Comparison ofR−1, R−1/2, ABC4, R1/2, ABC, and GA indices of Mr,s,t for fix t=1. Niagara azure, green, cyan, blue, purple, and red representsR−1, R−1/2, ABC4, R1/2, ABC, and GA indices, respectively.Figure 8
Comparison of first Zagreb coindex indexM1¯ and second Zagreb coindex M2¯ of Mr,s,t. Purple and blue represents M1¯ and M2¯, respectively.Figure 9
Comparison ofM1¯ and M2¯ indices of Mr,s,t, where purple color represents M1¯ and blue color represents M2¯.


## 5. Conclusion

In this article, we have studied the structureMr,s,t of M-carbon in the form of its degree and sum degree-based indices. The indices that we computed and discussed in the article are R−1, R−1/2, ABC4, R1/2, ABC, GA, GA5M1, R1,HM,S,M1¯, and M2¯. We also compared these indices numerically and graphically to show which type of indices behaved in different and same patterns (or values). We have found exact formulas for these indices for the structure of Mr,s,t of M-carbon.



---

*Source: 8792585-2021-10-18.xml*