# Valency-Based Topological Properties of Linear Hexagonal Chain and Hammer-Like Benzenoid

Authors: Yi-Xia Li; Abdul Rauf; Muhammad Naeem; Muhammad Ahsan Binyamin; Adnan Aslam
Journal: Complexity
 (2021)

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## Abstract

Topological indices are quantitative measurements that describe a molecule’s topology and are quantified from the molecule’s graphical representation. The significance of topological indices is linked to their use in QSPR/QSAR modelling as descriptors. Mathematical associations between a particular molecular or biological activity and one or several biochemical and/or molecular structural features are QSPRs (quantitative structure-property relationships) and QSARs (quantitative structure-activity relationships). In this paper, we give explicit expressions of two recently defined novel ev-degree- and ve-degree-based topological indices of two classes of benzenoid, namely, linear hexagonal chain and hammer-like benzenoid.


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## Body

## 1. Introduction

The combination of graph theory and chemistry is called chemical graph theory. An important use of graph theory is to mathematically model molecules. This helps the researchers to gain insight into the physical properties of these chemical compounds. Some physical properties such as the boiling point are related to the geometric structure of the compound. The chemical compounds have a variety of applications in pharmceutical chemistry. In chemistry, numerous compounds of different structures have the same chemical formula [1]. The chemical graph theory is a fast-rising field among researchers with applications in chemistry, biological, and electrical networks and drug designs [2–5].In the QSPR/QSAR analyses, scientists are interested in studying the topology of chemical networks linked to medicine, drug design, medical research, and experimental science using certain mathematical parameters obtained from the molecular structures of these networks [6]. Different properties or behaviors of chemical molecules have been investigated in the field of quantitative structure-property relationships [7, 8].Benzene is colorless and odour characteristic aromatic compound. Michael Faraday first isolated and identified benzene [9, 10]. Then, Eilhard Mitscherlich produced it by lime and benzoic acid [11], and Charles Mansfield from coal tar [12]. August Kekulé proposed the ring structure of six carbon atoms [13]. A big amount of benzene is used to manufacture cumene for phenol, and other amount is used in the manufacture of nylon fibers, detergents, drugs, and pesticides. Benzene is mobilized in the human body because of its exposure from tobacco smoke exhaust from motor vehicles and industries [14]. The compound having only benzene rings is known as the benzenoid aromatic compound.The topological descriptor concept was firstly given by Wiener [15] when working on paraffin, and the index number is correlated with the critical point, boiling points, and density [16, 17]. The topological indices are used to understand and develop mathematical characteristics of models by using edge, vertex, and degree concept. Mostly Randic, Wiener, and Zagreb indices are used in chemical and mathematical literature [18–21]. The Randic index variant is the harmonic index defined by Zhong [22]. However, all of the above research was performed by utilizing the classical concept of degrees. Chellali et al. introduced ve-degree and ev-degree [23], and Horoldagva et al. explored Chellali et al.’s work and gave mathematical concepts [24]. The degree-based ideology transformed into ve-degree and ev-degree.In this paper, we have discussed few ideologies of ve-degree and ev-degree. We present the ve-degree- and ev-degree-based topological indices for the molecular structure of multiple segment linear hexagonal chainL [m,n] and hammer like benzenoid Hn. We have presented ve-degree- and ev-degree-based topological indices for the molecular structure of for molecular structure of multiple segment linear hexagonal chains Lm,n and hammer-like benzenoid Hn. There is a wide-ranging research activity now on ve-degree- and ev-degree-based topological indices or variants. For further explanation of these topological indexes of different graphs and chemical structures, see [25–38].LetG be a graph with vertex set VG and edge set EG. For any vertex v∈VG, let γ(v) denotes the degree of v and is defined as the number of edges incident to v. The open neighborhood of v is denoted by N(v) and is defined as the set of vertices adjacent to v. The closed neighborhood of v, denoted by N[v] is defined as N[v] = N(v)∪v.The ev-degree of any edgeuv=e∈EG is the total number of the vertices of closed neighborhoods of the end vertices of an edge e, and the ev degree is denoted by Λeve.The ve-degree of any vertexv∈VG is the total number of different edges which is adjacent to v and the first neighbors of v, i.e., the sum of degrees of all closed neighborhood vertices of v.


## 2. Ev-Degree-Based Indices

For a connected graphG, the ev-degree-based Zagreb (Mev) index and Randic (Rev) index for any edge e=uv∈EG are defined as(1)MevG=∑e∈EΛeve2,RevG=∑e∈EΛeve−1/2.


### 2.1. Ve-Degree-Based Index

For a connected graphG, the ve-degree-based first Zagreb alpha (M1αve) index for any vertex v∈VG is defined as(2)M1αveG=∑v∈VΛvev2.


### 2.2. End Vertices’ Ve-Degree-Based Indices of Each Edge

For a connected graphG, the end vertices’ ve-degree-based indices for each edge, such as ve-degree-based first Zagreb beta index M1βve, second Zagreb beta index M2βve, atom-bond connectivity index ABCve [18], geometric-arithmetic index GAve [39], harmonic index Hve, sum-connectivity index χve [40], and the Randic index (Rve) for each edge uv∈EG, are defined as(3)M1βveG=∑uv∈EΛveu+Λvev,M2βveG=∑uv∈EΛveu×Λvev,ABCveG=∑uv∈EΛveu+Λvev−2Λveu×Λvev,GAveG=∑uv∈E2Λveu×ΛvevΛveu+Λvev,HveG=∑uv∈E2Λveu+Λvev,χveG=∑uv∈EΛveu+Λvev−1/2,RveG=∑uv∈EΛveu×Λvev−1/2.


## 2.1. Ve-Degree-Based Index

For a connected graphG, the ve-degree-based first Zagreb alpha (M1αve) index for any vertex v∈VG is defined as(2)M1αveG=∑v∈VΛvev2.


## 2.2. End Vertices’ Ve-Degree-Based Indices of Each Edge

For a connected graphG, the end vertices’ ve-degree-based indices for each edge, such as ve-degree-based first Zagreb beta index M1βve, second Zagreb beta index M2βve, atom-bond connectivity index ABCve [18], geometric-arithmetic index GAve [39], harmonic index Hve, sum-connectivity index χve [40], and the Randic index (Rve) for each edge uv∈EG, are defined as(3)M1βveG=∑uv∈EΛveu+Λvev,M2βveG=∑uv∈EΛveu×Λvev,ABCveG=∑uv∈EΛveu+Λvev−2Λveu×Λvev,GAveG=∑uv∈E2Λveu×ΛvevΛveu+Λvev,HveG=∑uv∈E2Λveu+Λvev,χveG=∑uv∈EΛveu+Λvev−1/2,RveG=∑uv∈EΛveu×Λvev−1/2.


## 3. Benzenoid

Benzene is an important parent aromatic compound of highly toxic nature having colorless and odour characteristics. Benzene is widely used in production volume such as to make synthetic fibers, dyes, plastic, resins, polystyrene, drugs, and pesticides and ranks in the top 20 chemicals [41]. The biggest application of benzene is in phenol preparation. Soon after its industrial use began, the toxicity of benzene was noticed in the blood forming organs [41]. In 1897, nine cases of benzene hematotoxicity were described [42] and further effects were documented in Weiskotten [43, 44] and Selling [45]. In 1928, the first case of benzene-related leukemia in a man was reported [41]. Benzene was first obtained from coal tar which was then replaced by processes based on petroleum. Chemist considered benzene puzzling in the 19th century that it could be so unreactive against additional reactions, considering its high degree of unsaturation. In 1865, finally, Kekulé proposed the ring structure [13]. We use commonly six carbon atoms that form a ring with an alternative single and double bond. The aromatic compound having only benzene rings in its form is known as the benzenoid aromatic compound. The multiple segment linear hexagonal chain of benzenoid is a prominent family of molecular benzenoid graphs, where this family formed on segment solely from benzenoid C6 (hexagon).


### 3.1. Crystallographic Structure of Multiple Segment Linear Hexagonal ChainLm,n

In this paper, we discuss two classes of benzenoid system, and first is a multiple segment linear hexagonal chains. A finite connected subgraph of the infinite hexagonal lattice without cut vertices or nonhexagonal interior faces is said to be a benzenoid system. In multiple segment linear hexagonal chains, six carbon atoms are connected through bond with angle 120° form hexagonal structure. Benzenoid system is called peri-condensed if three hexagons are connected by common vertex; otherwise, we say catacondensed.The acenes or polyacenes are a class of organic compounds and polycyclic aromatic hydrocarbons made up of linearly fused benzene rings. A linear chain of finite hexagonal structure rings of finite segments is called multiple segment linear hexagonal chainsLm,n. From Figure 1, we can obtain a single armchair chain for m=2 and n≥2. The crystal structure of Lm,n contains the 24mn−4n+3 vertices and 25mn−5n+3 edges, where m≥2 and n≥2. On the basis of degree of vertices, we divide it into two partitions in such way: 22mn−2n+3 vertices of degree 2 and 4mn−4n vertices of degree 3. Similarly, the edges of Lm,n are partitioned as E2,2 with 2n+5 edges, E2,3 with 8mn−12n+2 edges, and E3,3 having 2mn−1 edges.Figure 1
Structure of multiple segment linear hexagonal chainsLm,n.


## 3.1. Crystallographic Structure of Multiple Segment Linear Hexagonal ChainLm,n

In this paper, we discuss two classes of benzenoid system, and first is a multiple segment linear hexagonal chains. A finite connected subgraph of the infinite hexagonal lattice without cut vertices or nonhexagonal interior faces is said to be a benzenoid system. In multiple segment linear hexagonal chains, six carbon atoms are connected through bond with angle 120° form hexagonal structure. Benzenoid system is called peri-condensed if three hexagons are connected by common vertex; otherwise, we say catacondensed.The acenes or polyacenes are a class of organic compounds and polycyclic aromatic hydrocarbons made up of linearly fused benzene rings. A linear chain of finite hexagonal structure rings of finite segments is called multiple segment linear hexagonal chainsLm,n. From Figure 1, we can obtain a single armchair chain for m=2 and n≥2. The crystal structure of Lm,n contains the 24mn−4n+3 vertices and 25mn−5n+3 edges, where m≥2 and n≥2. On the basis of degree of vertices, we divide it into two partitions in such way: 22mn−2n+3 vertices of degree 2 and 4mn−4n vertices of degree 3. Similarly, the edges of Lm,n are partitioned as E2,2 with 2n+5 edges, E2,3 with 8mn−12n+2 edges, and E3,3 having 2mn−1 edges.Figure 1
Structure of multiple segment linear hexagonal chainsLm,n.


## 4. Main Results

Theorem 1.
LetHbe a molecular graph of multiple segment linear hexagonal chainsLm,n; then, ev-degree-based Zagreb index and ev-degree-based Randic index are given by(i)
MevH=272mn−268n+94.(ii)
RevH=8/5+2/6mn+1−12/5n+5/2+2/5−1/6.Proof.
From the structure of multiple segment linear hexagonal chainLm,n, there are three types of the edge partitions E2,2, E2,3, and E3,3 respectively, where the total number of the edges is 25mn−5n+3. By the definition of ev-degree, we have calculated the ev-degrees of the given edges partition as shown in Table 1.
From Table1, we have proved ev-degree-based indices such as(i)
The ev-degree-based Zagreb index:(4)MevH=∑e∈EHΛeve2MevH=42E2,2+52E2,3+62E3,3=422n+5+528mn−12n+2+622mn−1=162n+5+258mn−12n+2+362mn−1=272mn−268n+94.(ii)
The ev-degree-based Randic index:(5)RevH=∑e∈EHΛeve−1/2,RevH=4−1/2E4,2+5−1/2E2,3+6−1/2E3,3=4−1/22n+5+5−1/28mn−12n+2+6−1/22mn−1=85+26mn+1−125n+52+25−16.Table 1
The edges’ ev-degree ofLm,n.
Λu,ΛvΛeveFrequencyE2,242n+5E2,358mn−12n+2E3,362mn−1Theorem 2.
LetHbe a molecular graph of segment linear hexagonal chainLm,n; then, vertices’ ve-degree-based first Zagrebα-index is given by(6)M1αveH=340mn−324n+84.Proof.
From the structure of linear hexagonal chainLm,n, we divide the vertices into two partitions on the basis of degrees V2 and V3, respectively, where the total number of the vertices is 24mn−4n+3. By the definition of ve-degrees, we have calculated the vertices’ ve-degrees as shown in Table 2.Table 2
The vertices’ ve-degrees ofLm,n.
ΛuΛveuFrequency244254n+2264mn−8n374mn−8n+2384n−2By using above table, we have first ve-degree-based Zagrebα-index.(7)M1αveH=∑v∈VHΛvev2M1αveH=424+524n+2+624mn−8n+724mn−8n+2+824n−2=340mn−324n+84.Theorem 3.
LetHbe a molecular graph of linear hexagonal chainLm,n; then, end vertices’ ve-degree-based indices of each edges are given by(i)
M1βveH=134mn−134n+48(ii)
M2βveH=448mn−442n+81(iii)
ABCveH=811/42+213/56mn+28/5+42/7−2011/42+2−213/56+14/4n+6/2+27/5−8/5+22/7+211/42−14/8−1(iv)
GAveH=1642/13+456/15mn+235/3+4042/13+163/7−456/15+4n+410/7+165/9+35/3+442/13−83/7−2(v)
HveH=292/195mn+6491/5460n+46493/32760(vi)
χveH=8/13+2/15mn+2/10+2/3−20/13+4/14−2/15+1/2n+2/8+4/9−1/10+2/12+2/13−2/14−1/4(vii)
RveH=8/42+1/14mn+2/5+4/35−20/42+1/3−1/14+1/4n+7/40+2/5+2/35+2/42−1/23Proof.
From the structure of linear hexagonal chainLm,n, there are three types of edge partitions E2,2, E2,3, and E3,3, respectively, where the total number of the edges is 25mn−5n+3. According to definition of ve-degree of end vertices of each edge, we divide the new edges into five partitions, i.e., E1∗, E2∗, E3∗, E4∗, E5∗, E6∗, E7∗, and E8∗, respectively, as shown in Table 3.
By using the table, we can prove end vertices’ ve-degree-based indices of each edge such as(i)
The first Zagrebβ-index:(8)M1βveH=∑uv∈EHΛveu+Λvev,M1βveH=8E1∗+9E2∗+10E3∗+12E4∗+13E5∗+14E6∗+15E7∗+16E8∗=82+94+102n−1+124n+2+138mn−20n+2+144n−2+152mn−2n+162n−1=134mn−134n+48.(ii)
The second Zagreb beta index:(9)M2veH=∑uv∈EHΛveuΛvev,M2veH=16E1∗+20E2∗+25E3∗+35E4∗+42E5∗+48E6∗+56E7∗+64E8∗=162+204+252n−1+354n+2+428mn−20n+2+484n−2+562mn−2n+642n−1=448mn−442n+81.(iii)
The atom-bond connectivity index:(10)ABCveH=∑uv∈EHΛveu+Λvev−2ΛveuΛvev,ABCveH=616E1∗+720E2∗+825E3∗+1035E4∗+1142E5∗+1248E6∗+1356E7∗+1464E8∗=6162+7204+8252n−1+10354n+2+1142×8mn−20+2+12484n−2+13562mn−2n+14642n−1=81142+21356mn+285+427−201142+2−21356+144n+62+275−85+227+21142−148−1.(iv)
The geometric-arithmetic index:(11)GAveH=∑uv∈EH2Λveu×ΛvevΛveu+Λvev,GAveH=2107E1∗+2209E2∗+22510E3∗+23512E4∗+24213E5∗+24814E6∗+25615E7∗+26416E8∗=21072+22094+225102n−1+235124n+2+24213×8mn−20n+2+248144n−2+256152mn−2n+264162n−1=164213+45615mn+2353+404213+1637−45615+4n+4107+1659+353+44213−837−2.(v)
The harmonic index:(12)HveH=∑uv∈EH2Λveu+Λvev,HveH=28E1∗+29E2∗+210E3∗+212E4∗+213E5∗+214E6∗+215E7∗+216E8∗=282+294+2102n−1+2124n+2+2138mn−20n+2+2144n−2+2152mn−2n+2162n−1=292195mn+64915460n+4649332760.(vi)
The sum-connectivity index:(13)χveH=∑uv∈EHΛveu+Λvev−1/2,χveH=8−1/2E1∗+9−1/2E2∗+10−1/2E3∗+12−1/2E4∗+13−1/2E5∗+14−1/2E6∗+15−1/2E7∗+16−1/2E8∗=8−1/22+9−1/24+10−1/22n−1+12−1/24n+2+13−1/28mn−20n+2+14−1/24n−2+15−1/22mn−2n+16−1/22n−1=813+215mn+210+23−2013+414−215+12n+28+49−110+212+213−214−14.(vii)
The Randic index:(14)RveH=∑uv∈EHΛveuΛvev−1/2,RveH=16−1/2E1∗+20−1/2E2∗+25−1/2E3∗+35−1/2E4∗+42−1/2E5∗+48−1/2E6∗+56−1/2E7∗+64−1/2E8∗=16−1/22+20−1/24+25−1/22n−1+35−1/24n+2+42−1/28mn−20n+2+48−1/24n−2+56−1/22mn−2n+64−1/22n−1=842+114mn+25+435−2042+13−114+14n+740+25+235+242−123.Table 3
End vertices’ ve-degrees of each edge ofLm,n.
EdgeΛveu,ΛvevFrequencyE1∗(4, 4)2E2∗(4, 5)4E3∗(5, 5)2n−1E4∗(5, 7)4n+2for allnandm≥34for allnandm=2E5∗(6, 7)8mn−20n+2form≥30form=2E6∗(6, 8)4n−2for allnandm≥30for allnandm=2E7∗(7, 8)2mn−2nE8∗(8, 8)2n−1


### 4.1. Crystallographic Structure of Hammer-Like BenzenoidHn

Now, we discuss the second class of the benzenoid system that is a hammer-like structure. The linear polyacene of lengthn and both ends of polyacene are connected by pyrene fragments. Benzenoid system is called peri-condensed if three hexagons are connected by common vertex; otherwise, we say catacondensed. Hn is not a catacondensed benzenoid because a peri-condensed group is connected at end of polyacene, but in this paper, we discuss the structure of Hn as the varying fragments are catacondensed [46].We can obtain the hammer-like benzene structureHn by terminating the ends of a linear polyacene of length n with two pyrene fragments (see Figure 2). The crystal structure of Hn contains the 22n+15 vertices and 5n+39 edges. On the basis of degree of vertices, we divide it into two partitions in such way: 2n+8 vertices of degree 2 and 2n+7 vertices of degree 3. Similarly, the edges of Hn are partitioned as E2,2 with 10 edges, E2,3 with 4n+12 edges, and E3,3 having n+15 edges.Figure 2
Structure of hammer-like benzenoidHn.


## 4.1. Crystallographic Structure of Hammer-Like BenzenoidHn

Now, we discuss the second class of the benzenoid system that is a hammer-like structure. The linear polyacene of lengthn and both ends of polyacene are connected by pyrene fragments. Benzenoid system is called peri-condensed if three hexagons are connected by common vertex; otherwise, we say catacondensed. Hn is not a catacondensed benzenoid because a peri-condensed group is connected at end of polyacene, but in this paper, we discuss the structure of Hn as the varying fragments are catacondensed [46].We can obtain the hammer-like benzene structureHn by terminating the ends of a linear polyacene of length n with two pyrene fragments (see Figure 2). The crystal structure of Hn contains the 22n+15 vertices and 5n+39 edges. On the basis of degree of vertices, we divide it into two partitions in such way: 2n+8 vertices of degree 2 and 2n+7 vertices of degree 3. Similarly, the edges of Hn are partitioned as E2,2 with 10 edges, E2,3 with 4n+12 edges, and E3,3 having n+15 edges.Figure 2
Structure of hammer-like benzenoidHn.


## 5. Main Results

Theorem 4.
LetHbe a molecular graph of hammer-like structureHn; then, ev-degree-based Zagreb index and ev-degree-based Randic index are given by(i)
MevH=136n+1000.(ii)
RevH=4/5+1/6n+12/5+15/6+5.Proof.
From the structure of hammer-like structureHn, there are three types of the edge partitions E2,2, E2,3, and E3,3, respectively, where the total number of the edges is 5n+39. By the definition of ev-degree, we have calculated the ev-degrees of the given edges partition as shown in Table 4.
From Table4, we have proved ev-degree-based indices such as(i)
The ev-degree-based Zagreb index:(15)MevH=∑e∈EHΛeve2,MevH=42E2,2+52E2,3+62E3,3=4210+524n+12+62n+15=1610+254n+12+36n+15=136n+1000.(ii)
The ev-degree-based Randic index:(16)RevH=∑e∈EHΛeve−1/2,RevH=4−1/2E4,2+5−1/2E2,3+6−1/2E3,3=4−1/210+5−1/24n+12+6−1/2n+15=45+16n+125+156+5.Table 4
The edges’ ev-degree ofHn.
Λu,ΛvΛeveFrequencyE2,2410E2,354n+12E3,36n+15Theorem 5.
LetHbe a molecular graph of hammer-like structureHn; then, vertices’ ve-degree-based first Zagrebα-index is given by(17)M1αveH=170n+1298.Proof.
From the structure of hammer-like structureHn, we divide the vertices into two partitions on the basis of degrees V2 and V3, respectively, where the total number of the vertices is 22n+15. By the definition of ve-degrees, we have calculated the vertices’ ve-degrees as shown in Table 5.
By using Table5, we have first ve-degree-based Zagreb α-index:(18)M1αveH=∑v∈VHΛvev2,M1αveH=424+5212+622n+722n+2+828+924=170n+1298.Table 5
The vertices’ ve-degrees ofHn.
ΛuΛveuFrequency2442512262n372n+2388394Theorem 6.
LetHbe a molecular graph of hammer-like structureHn; then, end vertices’ ve-degree-based indices of each edge are given by(i)
M1βveH=66n+494(ii)
M2βveH=217n+1711(iii)
ABCveH=411/42+12/7n+87/20+42/5+8/3+411/42+45/24+314/4+8/9−411/42−12/7+414/63+2(iv)
GAveH=842/13+1n+1620/9+435/3+840/13+163/7+842/13+67/4+242/17+9(v)
HveH=69/91n+42001/7140(vi)
χveH=4/13+1/14n+8/9+2/10+8/12+4/14−1/14+3/2+4/17+2/18+1(vii)
RveH=4/42+1/7n+4/5+2/5+8/35+2/10+1/3−4/42−1/7+4/37+2/32+3/4+2/9Proof.
From the structure of hammer-like structureHn, there are three types of edge partitions E2,2, E2,3, and E3,3, respectively, where the total number of the edges is 5n+39. According to definition of ve-degree of end vertices of each edge, we divide the new edges into eleven partitions, i.e., E1∗, E2∗, E3∗, E4∗, E5∗, E6∗, E7∗, E8∗, E9∗, E10∗, and E11∗, respectively, as shown in Table 6.
By using the Table6, we can prove end vertices’ ve-degree-based indices of each edge such as(i)
The first Zagrebβ-index:(19)M1βveH=∑uv∈EHΛveu+Λvev,M1βveH=9E1∗+10E2∗+12E3∗+13E4∗+13E5∗+14E6∗+14E7∗+16E8∗+16E9∗+17E10∗+18E11∗=98+102+128+134+134n−4+144+14n−1+164+166+174+182=66n+494.(ii)
The second Zagreb beta index:(20)M2βveH=∑uv∈EHΛveuΛvev,M2βveH=20E1∗+25E2∗+35E3∗+40E4∗+42E5∗+48E6∗+49E7∗+63E8∗+64E9∗+72E10∗+81E11∗=208+252+358+404+424n−4+484+49n−1+634+646+724+812=217n+1711.(iii)
The atom-bond connectivity index:(21)ABCveH=∑uv∈EHΛveu+Λvev−2ΛveuΛvev,ABCveH=720E1∗+825E2∗+1035E3∗+1140E4∗+1142E5∗+1248E6∗+1249E7∗+1463E8∗+1464E9∗+1572E10∗+1681E11∗=7208+8252+10358+11404+11424n−4+12484+1249n−1+14634+14646+15724+16812=41142+127n+8720+425+83+41142+4524+89−41142+3144−127+41463+2.(iv)
The geometric-arithmetic index:(22)GAveH=∑uv∈EH2Λveu×ΛvevΛveu+Λvev,GAveH=2209E1∗+22510E2∗+23512E3∗+24013E4∗+24213E5∗+24814E6∗+24914E7∗+26316E8∗+26416E9∗+27217E10∗+28118E11∗=22098+225102+235128+240134+242134n−4+248144+24914n−1+263164+264166+272174+281182=84213+1n+16209+4353+84013+1637+84213+674+24217+9.(v)
The harmonic index:(23)HveH=∑uv∈EH2Λveu+Λvev,HveH=29E1∗+210E2∗+212E3∗+213E4∗+213E5∗+214E6∗+214E7∗+216E8∗+216E9∗+217E10∗+218E11∗=298+2102+2128+2134+2134n−4+2144+214n−1+2164+2166+2174+2182=6991n+420017140.(vi)
The sum-connectivity index:(24)χveH=∑uv∈EHΛveu+Λvev−1/2,χveH=9−1/2E1∗+10−1/2E2∗+12−1/2E3∗+13−1/2E4∗+13−1/2E5∗+14−1/2E6∗+14−1/2E7∗+16−1/2E8∗+16−1/2E9∗+17−1/2E10∗+18−1/2E11∗=198+1102+1128+1134+1134n−4+1144+114n−1+1164+1166+1174+1182=413+114n+89+210+812+414−114+32+417+218+1.(vii)
The Randic index:(25)RveH=∑uv∈EHΛveuΛvev−1/2,RveH=20−1/2E1∗+25−1/2E2∗+35−1/2E3∗+40−1/2E4∗+42−1/2E5∗+48−1/2E6∗+49−1/2E7∗+63−1/2E8∗+64−1/2E9∗+72−1/2E10∗+81−1/2E11∗=1208+1252+1358+1404+1424n−4+1484+149n−1+1634+1646+1724+1812=442+17n+45+25+835+210+13−442−17+437+232+34+29.Table 6
End vertices’ ve-degrees of each edge ofHn.
EdgeΛveu,ΛvevFrequencyE1∗(4, 5)8E2∗(5, 5)2E3∗(5, 7)8E4∗(5, 8)4E5∗(6, 7)4n−4E6∗(6, 8)4E7∗(7, 7)n−1E8∗(7, 9)4E9∗(8, 8)6E10∗(8, 9)4E11∗(9, 9)2


## 6. Numerical Results and Discussion

The experimental studies have a broad range of applications in the fields of chem-informatics, bio-informatics, and bio-medicine, where various graph topological-based assessments are used to tackle many complicated schemes. We have carried out the calculations of the numerical results depending upon the topological indices related to crystallographic molecular structure of multiple segment linear hexagonal chainLm,n and hammer-like benzenoid Hn. We have computed the ev-degree, ve-degree, and end-vertex degree-based topological indices, namely, Mev, M1αve, M1βve, M2βve, Rve, Rev, (ve-ABC), (ve-GA), (ve-H), and (ve-χ). In order to see the physical interpretation of the calculated indices, we have illustrated the results with the help of graphical representations and numerical tables. The results are shown in Figures 3(a)–8 and Tables 7 and 8 for the crystallographic molecular structure of multiple segment linear hexagonal chain Lm,n and in the Figure 9 and Tables 9 and 10 for the crystallographic molecular structure of hammer-like benzenoid Hn. It can be seen from Tables 7–10 and the Figures 3–9 that with an increase in m and n, all the computed topological descriptors gave an increased value.Figure 3
The ev-degree-based indices. (a) The Zagreb index. (b) The Randic index.
(a)(b)Figure 4
The first Zagrebα-index.Figure 5
(a) The first Zagrebβ-index (I). (b) The second Zagreb index.
(a)(b)Figure 6
(a) The atom-bond connectivity index. (b) The geometric-arithmetic index.
(a)(b)Figure 7
(a) The harmonic Index. (b) The sum-connectivity index.
(a)(b)Figure 8
The Randic index.Table 7
Numerical representation of the computed indices for certain values ofm and n.
m,nMevHRevHM1αveHM1βveHM2veH1,1983.013810048872,264611.829911203169893,3173829.4343282085227874,4337455.82725200165654815,5555491.00858260272890716,68278134.9782120004068135577,711546187.7363164205676189398,815358249.2828215207552252179,919714319.61772730096963239110,1024614398.7411337601210840461Table 8
Numerical representation of the computed indices for certain values ofm and n.
m,nABCveHGAveHHveHχveHRveH1,13.71400.04558894.10552.09971.47932,213.89320.10535279.78667.59814.53453,334.18790.1850618.462618.566910.59304,464.59810.284711330.133535.006119.65505,5105.12390.40430644.799256.915731.72036,6155.76510.54384562.459984.295746.78917,7216.52180.703326983.1154117.146164.86128,8287.39410.8827528106.7657155.466985.93669,9368.38181.0821225133.4110199.2580110.015510,10459.48511.301436163.0511248.5196137.0977Figure 9
(a)Mev, M1αve, M1βve, and M2βve. (b) ABCve, GAve, Hve, and χve. (c) Rev and Rve.
(a)(b)(c)Table 9
Numerical representation of the computed indices for certain values ofn.
nMevHRevHM1αveHM1βveHM2βveH[1]113618.687414685601928[2]127220.884516386262145[3]140823.081618086922362[4]154425.278719787582579[5]168027.475821488242796[6]181629.672923188903013[7]195231.870024889563230[8]208834.0671265810223447[9]222436.2642282810883664[10]236038.4613299811543881Table 10
Numerical representation of the computed indices for certain values ofn.
nABCveHGAveHHveHχveH)RveH[1]21.936647.63106.64079.95076.6985[2]24.478552.61927.399011.32747.4586[3]27.020557.60738.157212.70418.2186[4]29.562462.59558.915514.08078.9787[5]32.104367.58369.673715.45749.7388[6]34.646372.571810.431916.834010.4988[7]37.188277.559911.190218.210711.2589[8]39.730282.548111.948419.587412.0190[9]42.272187.536212.706720.964012.7791[10]44.814092.524413.464922.340713.5391The Zagreb type indices helped in computing totalπ-electron energy of molecules [47]; thus, the total π-electron energy is in increasing order for the case of multiple segment linear hexagonal chain Lm,n and the hammer-like benzenoid Hn, for higher values of m and n. The Randic index can be used in the study of the chemical similarity of molecular compounds [48] . Moreover, the Randic index can be used in computing the Kovats constants and boiling point of molecules. The value of randic index increases with increase in the values of m and n for multiple segment linear hexagonal chain L[m, n] and the hammer like benzenoid Hn. Moreover, for both structures the GA index also increases with the increase in value of m and n. Moreover, GA index for the multiple segment linear hexagonal chains Lm,n and the hammer-like benzenoid Hn increases with the increment in m and n. The atom-bond connectivity (ABC) index provides a very good correlation for computing the strain energy of cycloalkanes as well as for the stability of linear and branched alkanes [49]. The ABC index increases for Lm,n and Hn with the increase in the values of m and n.


## 7. Conclusion

The topological index (TI) is a quantitative measure that represents the physical, biological, and chemical behavior of essential organic molecules such as moisture and forming heat; boiling, melting, and flickering point; and temperature, strain, partition coefficient, fluid voltage, density, and molecular weight in chromatography. Graph topologies are important tools for approximating and predicting the properties of biological and chemical compounds in the analysis of the quantitative structure-property relationships (QSPRs) and the quantitative structure-activity relationships (QSARs). In this paper, we have evaluated the ev- and ve-degree-based topological indices with graphical representations for molecular structure of multiple segment linear hexagonal chainsLm,n and the hammer-like benzenoid Hn to study the pharmaceutical and chemical properties and biological behavior. Further studies are also required for the better understanding of multiple segment linear hexagonal chain Lm,n and the hammer-like benzenoid Hn.



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