# Topological Indices of the Pent-Heptagonal Nanosheets VC5C7 and HC5C7

Authors: Fei Deng; Xiujun Zhang; Mehdi Alaeiyan; Abid Mehboob; Mohammad Reza Farahani
Journal: Advances in Materials Science and Engineering
 (2019)

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## Abstract

In this paper, we computed the topological indices of pent-heptagonal nanosheet. Formulas for atom-bond connectivity index, fourth atom-bond connectivity index, Randić connectivity index, sum-connectivity index, first Zagreb index, second Zagreb index, augmented Zagreb index, modified Zagreb index, hyper Zagreb index, geometric-arithmetic index, fifth geometric-arithmetic index, Sanskruti index, forgotten index, and harmonic index of pent-heptagonal nanosheet have been derived.


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## Body

## 1. Introduction

Nanostructures, which have scale less than 100 nm, include nanotubes, nanosheets, and nanoparticles. Nanosheets (two-dimensional nanostructures) have large surface and sharp edge, which make them play an important role in many applications such as energy storage [1–3], optoelectronics [4], catalysis [5], and bioelectronics [6]. Graphene [7], silicone [8], and borophene [9] are typical nanosheets. Graphene is the basic nanosheet that exhibits the honeycomb lattice structure. Because of the unique structural, electrical, optical, and mechanical properties, graphene nanosheets drew dramatic attention of academic and industrial research [10].People found some defects in graphene nanosheets made through experiments. In Figure1, the 5–7 defect in graphene is illustrated. As nanoscale-level defects introduced into graphene could be extremely useful and exploited to generate novel, innovative, and useful materials and devices [10], the 5–8–5 [11, 12] and 5–7–7–5 defects are obtained by several approaches [13–16]. Naturally, the property of C5C7 nanostructures has become an active area of research [17–22].Figure 1
The C5C7 defect.In chemical graph theory, topological indices are very useful to describe molecules or nanostructures. The vertices of the graph correspond to the atoms of molecules, and the edges correspond to chemical bonds.In this paper, we only consider undirected graphs without loops and multiple edges. LetG be a graph. We denote V(G) and E(G) as the vertex set and edge set of G, respectively. We denote by dv the degree of the vertex v of graph G and Nv the set of neighbors of v. We denote by Ea,b the set of edges with degrees of end vertices a and b, i.e.,(1)Ea,b=uva,b=du,dv.Let the sum of degrees of neighboring vertices ofu be denoted by sGu, which is defined as(2)sGu=∑v∈Nudv.We define(3)Ea,b′=uva,b=sGu,sGv.In 1998, Estrada et al. [23] defined the atom-bond connectivity index as(4)ABCG=∑e=uv∈EGdu+dv−2dudv.The ABC4 index is defined as(5)ABC4G=∑e=uv∈EGsGu+sGv−2sGusGv.The Randić connectivity index (see [24]) is defined on the ground of vertex degrees as(6)RG=∑e=uv∈EG1dudv.The sum-connectivity index is defined (see [25]) as(7)XG=∑e=uv∈EG1du+dv.The first Zagreb indexM1G is defined as the sum of squares of the vertex degrees du and dv of vertices u and v in graph G (see [26]). The mathematical formula of first Zagreb index is(8)M1G=∑v∈VGdv2=∑e=uv∈EGdu+dv.The second Zagreb indexM2G is defined as the sum of squares of the vertex degrees du and dv of vertices u and v in graph G (see [26]). The mathematical formula of second Zagreb index is(9)M2G=∑e=uv∈EGdudv.Augmented Zagreb index AZI(G) (see [27]) is defined as(10)AZIG=∑e=uv∈EGdudvdu+dv−23.Modified Zagreb indexM2∗G (see [28]) is defined as(11)M2∗G=∑e=uv∈EG1dudv.In 2013, Shirdel et al. [28] introduced hyper Zagreb index as(12)HMG=∑e=uv∈EGdudv2.The geometric-arithmetic index (see [29]) is defined as(13)GAG=∑e=uv∈EG2dudvdu+dv.TheGA5 is Defined as(14)GA5G=∑e=uv∈EG2sGusGvsGu+sGv.For a detailed survey about topological indices and applications of topological indices, we refer [30–38] and references therein. Recently, motivated by the previous research on topological descriptors and their applications, Hosamani [39] proposed the Sanskruti index which can be utilized to guess the bioactivity of chemical compounds and shows good correlation with entropy of octane isomers.The Sanskruti index of graphG is defined as(15)SG=∑uv∈EGsGusGvsGu+sGv−23.The forgotten indexFG is defined as the sum of cubic of the vertex degrees in G. Mathematical formula of forgotten index is [40](16)FG=∑v∈VGdv3=∑uv∈EGdu2+dv2.The harmonic indexHG (see [41]) is defined as(17)HG=∑uv∈EG2du+dv.


## 2. C5C7 Nanosheets and Their Structures

A C5C7 net is a trivalent decoration made by alternating C5 and C7. It can cover either a cylinder or a torus and certainly able to cover a sheet.The topological indices of nanostructures have been studied extensively. Graovac et al. [42] discussed the GA index of TUC4C8(S) nanotube in 2009. Graovac et al. [42] computed fifth geometric-arithmetic index for nanostar dendrimers in 2011. Asadpour et al. [43] computed Randić, Zagreb, and ABC indices of TUC4C8(S) and TUC4C8(R) nanotubes and V-phenylenic nanotorus in 2011. Al-Fozan et al. [44] computed the Szeged index of C4C8(S) and H-naphtalenic nanosheets (2n, 2m) in 2014. Azari and Iranmanesh [45] computed in Harary index of Pm × Pn (nanosheets), C4 nanotubes, and C4 nanotori in 2014. Ashrafi and Loghman [46] discussed the Padmakar–Ivan (PI) index of TUC4C8(S) nanotubes in 2006.The topological indices of C5C7 nanostructures have also been studied. Iranmanesh et al. [47] introduced a GAP program to compute the Szeged and PI indices of VC5C7[p,q] and HC5C7[p,q] nanotubes in 2008. Farahani and Zhang [48–53] discussed Randić connectivity index and sum connectivity index of the pent-heptagonal nanotube VAC5C7(S) in 2013. For the nanosheet VC5C7(m, n), we denote the number of pentagons in the first row by n; in this nanotube, the first four rows of vertices and edges are repeated alternatively and we denote the number of this repetition by m (see, e.g.,Figure 2(a) with (m, n) = (2, 4)).For the nanosheet HC5C7(m, n), we denote the number of heptagons in the first row by n. In this nanotube, the first four rows of vertices and edges are repeated alternatively and we denote the number of this repetition by m (see, e.g., Figure 2(b) with (m, n) = (2, 4)).Figure 2
The nanosheets (a) VC5C7(2, 4) and (b) HC5C7(2, 4).
(a)
(b)The nanosheet VC5C7(2, 4) has 16mn + 2m + 5n vertices and 24mn + 4n edges. Moreover, it has 6m + 7n vertices with degree 2 and 16mn − 4m − 2n vertices with degree 3. The edge partitions according to different Es,t and Es,t′ are presented in Tables 1 and 2, respectively. We computed some topological indices according to different types of Es,t and Es,t′. As an example, Figures 3(a) and 3(b) show the edge partition of the nanosheet VC5C7(2, 4) by different Es,t and Es,t′, respectively.Table 1
Partition of the edge set of VC5C7(m, n) into Es,t.
(s, t)
(2, 3)
(3, 3)
(2, 2)
E
s
,
t
′
8m + 10n − 8
24mn − 10m − 8n + 4
2m + 2n + 4Table 2
Partition of the edge set of VC5C7(m, n) into Es,t′.
(s, t)
(6, 7)
(7, 9)
(5, 7)
E
s
,
t
′
4m + 6n − 8
4m + 4n − 3
4m + 2n + 2
(s, t)
(9, 9)
(4, 5)
(5, 5)
E
s
,
t
′
24mn − 14m − 15n + 10
8
2m + 2n − 4
(s, t)
(8, 9)
(5, 8)
(8, 8)
E
s
,
t
′
2n − 2
2n − 2
n − 2Figure 3
The edge partition of the nanosheets (a) VC5C7(2, 4) by different Es,t and (b) VC5C7(2, 4) by different Es,t′.
(a)
(b)The nanosheet HC5C7(2, 4) has 16mn + 2m + 4n vertices and 24mn + 3n edges. Moreover, it has 6m + 6n vertices with degree 2 and 16mn − 4m − 2n vertices with degree 3. We computed some topological indices according to different types of Es,t and Es,t′. The edge partitions according to different Es,t and Es,t′ are presented in Tables 3 and 4, respectively. As an example, Figures 4(a) and 4(b) show the edge partition of the nanosheet HC5C7(2, 4) by different Es,t and Es,t′, respectively.Table 3
Partition of the edge set of HC5C7(m, n) into Es,t.
(s, t)
(3, 3)
(2, 3)
(2, 2)
E
s
,
t
′
24mn − 10m − 6n + 10
8m + 6n − 4
2m + 3n + 2Table 4
Partition of the edge set of HC5C7(m, n) into Es,t′.
(s, t)
(8, 9)
(6, 8)
(8, 8)
E
s
,
t
′
8m + 6n − 12
4m + 2n − 4
4m + n
(s, t)
(9, 9)
(7, 9)
(6, 7)
E
s
,
t
′
24mn − 22m − 14n + 13
n + 1
2n
(s, t)
(5, 7)
(4, 4)
—
E
s
,
t
′
2
n
—Figure 4
The edge partition of the nanosheets (a) HC5C7(2, 4) by different Es,t (b) and HC5C7(2, 4) by different Es,t′.
(a)
(b)


## 3. Main Results

Theorem 1.
(i)
LetG be a VC5C7(m, n) nanosheet, then(18)ABCG=16mn+62−163n+52−203m−22+83.(ii)
LetG be a HC5C7(m, n) nanosheet, then(19)ABCG=52m−4n−20m3+922n+16mn−2+43.Proof.
(i)
The proof is calculated based on the edge partition given in Table1. By the definition of ABC index, we have(20)ABCG=E2,336+E3,349+E2,224=8m+10n−836+24mn−10m−8n+449+2m+2n+424n=16mn+62−163n+52−203m−22+83.(ii)
The proof is calculated based on the edge partition given in Table3. By the definition of ABC index, we have(21)ABCG=E3,349+E2,336+E2,224=24mn−10m−6n+249+8m+6n−436+2m+3n+224n=52m−4n−20m3+922n+16mn−2+43.Theorem 2.
(i)
LetG be a VC5C7(m, n) nanosheet, then(22)ABC4G=322m15−20n3−56m9+414m7+2462m21+322n15+2314n56+30n6+110n10+462n7+32mn3−1325+91456−306+4355−11010−446221+409.(ii)
LetG be a HC5C7(m, n) nanosheet, then(23)ABC4G=2147−47n9−70m9+32mn3+2n+13+222m−25+1102m+n−110+304m+3n−66+35n+25+6n4+462n21+144m+n8+349.Proof.
(i)
The proof is calculated based on the edge partition given in Table2. By the definition of ABC4 index, we have(24)ABC4G=1142E6,7′+1463E7,9′+1035E5,7′+1681E9,9′+720E4,5′+825E5,5′+1572E8,9′+1140E5,8′+1464E8,8′=11424m+6n−8+14634m+4n−3+10354m+2n+2+168124mn−14m−15n+10+7208+8252m+2n−4+15722n−2+11402n−2+1464n−1=322m15−20n3−56m9+414m7+2462m21+322n15+2314n56+30n6+110n10+462n7+32mn3−1325+91456−306+4355−11010−446221+409.(ii)
The proof is calculated based on the edge partition given in Table4. By the definition of ABC4 index, we have(25)ABC4G=1572E8,9′+1248E6,8′+1464E8,8′+1681E9,9′+1463E7,9′+1142E6,7′+1140E5,8′+720E4,5′+825E5,5′+1035E5,7′+616E4,4′=15728m+6n−12+12484m+2n−4+14644m+n+168124mn−22m−14n+13+1463n+1+11422n+11404m+2n−2+7202n+4+8252m−2+1035×2+616n=2147−47n9−70m9+32mn3+2n+13+222m−25+1102m+n−110+304m+3n−66+35n+25+6n4+462n21+144m+n8+349.Theorem 3.
(i)
LetG be a VC5C7(m, n) nanosheet, then(26)RG=8mn−5n3−7m2+668m+10n−8+103.(ii)
LetG be a HC5C7(m, n) nanosheet, then(27)RG=8mn−n2−7m3+634m+3n−2+53.Proof.
(i)
The proof is calculated based on the edge partition given in Table1. By the definition of the Randić connectivity index, we have(28)RG=E2,316+E3,319+E2,214=8m+10n−816+24mn−10m−8n+419+2m+2n+414=8mn−5n3−7m3+634m+5n−4+103.(ii)
The proof is calculated based on the edge partition given in Table3. By the definition of the Randić connectivity index, we have(29)RG=E3,319+E2,316+E2,214=24mn−10m−6n+219+8m+6n−416+2m+3n+214=8mn−n2−7m3+634m+3n−2+53.Theorem 4.
(i)
LetG be a VC5C7(m, n) nanosheet, then(30)XG=m+n+58m+10n−85−65m+4n−12mn−23+2.(ii)
LetG be a HC5C7(m, n) nanosheet, then(31)XG=m+3n2+58m+6n−45−65m+3n−12mn−13+1.Proof.
(i)
The proof is calculated based on the edge partition given in Table1. By the definition of the sum connectivity index, we have(32)XG=E2,315+E3,316+E2,214=8m+10n−815+24mn−10m−8n+416+2m+2n+414=m+n+58m+10n−85−65m+4n−12mn−23+2.(ii)
The proof is calculated based on the edge partition given in Table3. By the definition of the sum connectivity index, we have(33)XG=E3,316+E2,315+E2,214=24mn−10m−6n+216+8m+6n−415+2m+3n+214=m+3n2+58m+6n−45−65m+3n−12mn−13+1.Theorem 5.
(i)
LetG be a VC5C7(m, n) nanosheet, then(34)M1G=10n−12m+144mn.(ii)
LetG be a HC5C7(m, n) nanosheet, then(35)M1G=6n−12m+144n.Proof.
(i)
The proof is calculated based on the edge partition given in Table1. By the definition of the first Zagreb index, we have(36)M1G=5E2,3+6E3,3+4E2,2=58m+10n−8+624mn−10m−8n+4+42m+2n+4=10n−12m+144mn.(ii)
The proof is calculated based on the edge partition given in Table3. By the definition of the first Zagreb index, we have(37)M1G=6E3,3+5E2,3+4E2,2=624mn−10m−6n+2+58m+6n−4+42m+3n+2=6n−12m+144n.Theorem 6.
(i)
LetG be a VC5C7(m, n) nanosheet, then(38)M2G=216mn−4n−34m+4.(ii)
LetG be a HC5C7(m, n) nanosheet, then(39)M2G=216mn−6n−34m+2.Proof.
(i)
The proof is calculated based on the edge partition given in Table1. By the definition of the second Zagreb index, we have(40)M2G=6E2,3+9E3,3+4E2,2=68m+10n−8+924mn−10m−8n+4+42m+2n+4=216mn−4n−34m+4.(ii)
The proof is calculated based on the edge partition given in Table3. By the definition of the second Zagreb index, we have(41)M2G=9E3,3+6E2,3+4E2,2=924mn−10m−6n+2+68m+6n−4+42m+3n+2=216mn−6n−34m+2.Theorem 7.
(i)
LetG be a VC5C7(m, n) nanosheet, then(42)AZIG=39n8−1085m32+2187mn8+21716.(ii)
Let G be a HC5C7(m, n) nanosheet, then(43)AZIG=117n32−1085m32+2187mn8+21732.Proof.
(i)
The proof is calculated based on the edge partition given in Table1. By the definition of the augmented Zagreb index, we have(44)AZIG=633E2,3+943E3,3+423E2,2=6338m+10n−8+94324mn−10m−8n+4+4232m+2n+4=39n8−1085m32+2187mn8+21716.(ii)
The proof is calculated based on the edge partition given in Table3. By the definition of the augmented Zagreb index, we have(45)AZIG=943E3,3+633E2,3+423E2,2=94324mn−10m−6n+2+6338m+6n−4+4232m+3n+2=117n32−1085m32+2187mn8+21732.Theorem 8.
(i)
LetG be a VC5C7(m, n) nanosheet, then(46)M2∗G=13m18+23n18+8mn3+19.(ii)
LetG be a HC5C7(m, n) nanosheet, then(47)M2∗G=13m18+13n12+8mn3+118.Proof.
(i)
The proof is calculated based on the edge partition given in Table1. By the definition of the modified Zagreb index, we have(48)M2∗G=12×3E2,3+13×3E3,3+12×2E2,2=12×38m+10n−8+13×324mn−10m−8n+4+12×22m+2n+4=13m18+23n18+8mn3+19.(ii)
The proof is calculated based on the edge partition given in Table3. By the definition of the modified Zagreb index, we have(49)M2∗G=13×3E3,3+12×3E2,3+12×2E2,2=13×324mn−10m−6n+2+12×38m+6n−4+12×22m+3n+2=13m18+13n12+8mn3+118.Theorem 9.
(i)
LetG be a VC5C7(m, n) nanosheet, then(50)HMG=864mn−6n−128m+8.(ii)
LetG be a HC5C7(m, n) nanosheet, then(51)HMG=864mn−18n−128m+4.Proof.
(i)
The proof is calculated based on the edge partition given in Table1. By the definition of the hyper Zagreb index, we have(52)HMG=2+32E2,3+3+32E3,3+2+22E2,2=2+328m+10n−8+3+3224mn−10m−8n+4+2+222m+2n+4=864mn−6n−128m+8.(ii)
The proof is calculated based on the edge partition given in Table3. By the definition of the hyper Zagreb index, we have(53)HMG=3+32E3,3+2+32E2,3+2+22E2,2=3+3224mn−10m−6n+2+2+328m+6n−4+2+222m+3n+2=864mn−18n−128m+4.Theorem 10.
(i)
LetG be a VC5C7(m, n) nanosheet, then(54)GAG=4m+4n+3016m+20n−165−610m+8n−24mn−4+8.(ii)
LetG be a HC5C7(m, n) nanosheet, then(55)GAG=4m+6n+3016m+12n−85−610m+6n−24mn−2+4.Proof.
(i)
The proof is calculated based on the edge partition given in Table1. By the definition of the first geometric-arithmetic index, we have(56)GAG=2×65E2,3+2×96E3,3+2×44E2,2=2×658m+10n−8+2×9624mn−10m−8n+4+2×442m+2n+4=4m+4n+3016m+20n−165−610m+8n−24mn−4+8.(ii)
The proof is calculated based on the edge partition given in Table3. By the definition of the first geometric-arithmetic index, we have(57)GAG=2×96E3,3+2×65E2,3+2×44E2,2=2×9624mn−10m−6n+2+2×658m+6n−4+2×442m+3n+2=4m+6n+3016m+12n−85−610m+6n−24mn−2+4.Theorem 11.
(i)
LetG be a VC5C7(m, n) nanosheet, then(58)GA5G=4n+3253+374m+4n−32−3214m+15n−24mn−10+104m+4n−82+1054m+2n+23+5468m+12n−1613+6344n−417+21304n−413−4.(ii)
LetG be a HC5C7(m, n) nanosheet, then(59)GA5G=16m+4n+21053+22n+37n+12+254n+83−3222m+14n−24mn−13+2428m+4n−87+21308m+4n−413+63416m+12n−2417+102m−2+4546n13.Proof.
(i)
The proof is calculated based on the edge partition given in Table2. By the definition of the fifth geometric-arithmetic index, we have(60)GA5G=2×4213E6,7′+2×6316E7,9′+2×3512E5,7′+2×8118E9,9′+2×209E4,5′+2×2510E5,5′+2×7217E8,9′+2×4013E5,8′+2×6416E8,8′=2×42134m+6n−8+2×63164m+4n−3+2×35124m+2n+2+2×811824mn−14m−15n+10+2×2098+2×25102m+2n−4+2×72172n−2+2×40132n−2+2×6416n−1=4n+3253+374m+4n−32−3214m+15n−24mn−10+104m+4n−82+1054m+2n+23+5468m+12n−1613+6344n−417+21304n−413−4.(ii)
The proof is calculated based on the edge partition given in Table4. By the definition of the fifth geometric-arithmetic index, we have(61)GA5G=2×7217E8,9′+2×4814E6,8′+2×6416E8,8′+2×8118E9,9′+2×6316E7,9′+2×4213E6,7′+2×4013E5,8′+2×209E4,5′+2×2510E5,5′+2×3512E5,7′+2×168E4,4′=2×72178m+6n−12+2×48144m+2n−4+2×64164m+n+2×811824mn−22m−14n+13+2×6316n+1+2×42132n+2×40134m+2n−2+2×2092n+4+2×25102m−2+2×3512×2+2×168n=16m+4n+21053+22n+37n+12+254n+83−3222m+14n−24mn−13+2428m+4n−87+21308m+4n−413+63416m+12n−2417+102m−2+4546n13.Theorem 12.
(i)
LetG be a VC5C7(m, n) nanosheet, then(62)SG=1594323mn512−14620607759501n21249536000−2717055933m2725888+36950846358693116872448000.(ii)
LetG be a HC5C7(m, n) nanosheet, then(63)SG=1594323mn512−1783308545665133n3155556096000−126009252864147m116872448000+50620823728941233744896000.Proof.
(i)
The proof is calculated based on the edge partition given in Table2. By the definition of the Sanskruti index, we have(64)SG=42113E6,7′+63143E7,9′+35103E5,7′+81163E9,9′+2073E4,5′+2583E5,5′+72153E8,9′+40113E5,8′+64143E8,8′=421134m+6n−8+631434m+4n−3+351034m+2n+2+8116324mn−14m−15n+10+2073×8+25832m+2n−4+721532n−2+401132n−2+64143n−1=1594323mn512−14620607759501n21249536000−2717055933m2725888+36950846358693116872448000.(ii)
The proof is calculated based on the edge partition given in Table4. By the definition of the Sanskruti index, we have(65)SG=72153E8,9′+48123E6,8′+64143E8,8′+81163E9,9′+63143E7,9′+42113E6,7′+40113E5,8′+2073E4,5′+2583E5,5′+35103E5,7′+1663E4,4′=721538m+6n−12+481234m+2n−4+641434m+n+8116324mn−22m−14n+13+63143n+1+421132n+401134m+2n−2+20732n+4+25832m−2+35103×2+1663n=1594323mn512−1783308545665133n3155556096000−126009252864147m116872448000+50620823728941233744896000.Theorem 13.
(i)
LetG be a VC5C7(m, n) nanosheet, then(66)FG=432mn+2n−60m.(ii)
LetG be a HC5C7(m, n) nanosheet, then(67)FG=432mn−6n−60m.Proof.
(i)
The proof is calculated based on the edge partition given in Table1. By the definition of the forgotten topological index, we have(68)FG=22+32E2,3+32+32E3,3+22+22E2,2=22+328m+10n−8+32+3224mn−10m−8n+4+22+222m+2n+4=432mn+2n−60m.(ii)
The proof is calculated based on the edge partition given in Table3. By the definition of the forgotten topological index, we have(69)FG=32+32E3,3+22+32E2,3+22+22E2,2=32+3224mn−10m−6n+2+22+328m+6n−4+22+222m+3n+2=432mn−6n−60m.Theorem 14.
(i)
LetG be a VC5C7(m, n) nanosheet, then(70)HG=13m15+7n3+8mn+215.(ii)
LetG be a HC5C7(m, n) nanosheet, then(71)HG=13m15+19n10+8mn+115.Proof.
(i)
The proof is calculated based on the edge partition given in Table1. By the definition of the harmonic index, we have(72)HG=22+3E2,3+23+3E3,3+22+2E2,2=22+38m+10n−8+23+324mn−10m−8n+4+22+22m+2n+4=13m15+7n3+8mn+215.(ii)
The proof is calculated based on the edge partition given in Table3. By the definition of the harmonic index, we have(73)HG=23+3E3,3+22+3E2,3+22+2E2,2=23+324mn−10m−6n+2+22+38m+6n−4+22+22m+3n+2=13m15+19n10+8mn+115.


## 4. Conclusions

QSARs represent predictive models derived from the application of statistical tools correlating to the biological activity (including desirable therapeutic effects and undesirable side effects) of chemicals (drugs/toxicants/environmental pollutants) with descriptors representative of molecular structure and/or properties. QSARs are being applied in many disciplines like risk assessment, toxicity prediction, and regulatory decisions in addition to drug discovery and lead optimization. In this paper, some important topological indices of the pent-heptagonal nanosheet that is covered by pentagons and heptagons are calculated, and formulas for their atom-bond connectivity index, fourth atom-bond connectivity index, Randić connectivity index, sum-connectivity index, first Zagreb index, second Zagreb index, augmented Zagreb index, modified Zagreb index, hyper Zagreb index, geometric-arithmetic index, fifth geometric-arithmetic index, Sanskruti index, forgotten index, and harmonic index of two types of pent-heptagonal nanosheets are derived. Our calculated results help in understanding topology of understudy nanotubes. For instance, the first Zagreb index is used to be known as pi-electronic energy. The augmented Zagreb index can be used to investigate the heat of formation. Randić index is the mostly applied and investigated degree-based topological index.



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*Source: 9594549-2019-06-23.xml*