# Computation of Topological Indices of Double and Strong Double Graphs of Circumcoronene Series of BenzenoidHm

Authors: Muhammad Shoaib Sardar; Imran Siddique; Dalal Alrowaili; Muhammad Asad Ali; Shehnaz Akhtar
Journal: Journal of Mathematics
 (2022)

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## Abstract

Topological indices are very useful to assume certain physiochemical properties of the chemical compound. A molecular descriptor which changes the molecular structures into certain real numbers is said to be a topological index. In chemical graph theory, to create quantitative structure activity relationships in which properties of molecule may be linked with their chemical structures relies greatly on topological indices. The benzene molecule is a common chemical shape in chemistry, physics, and nanoscience. This molecule could be very beneficial to synthesize fragrant compounds. The circumcoronene collection of benzenoidHm is one family that generates from benzene molecules. The purpose of this study is to calculate the topological indices of the double and strong double graphs of the circumcoronene series of benzenoids Hm. In addition, we also present a numerical and graphical comparison of topological indices of the double and strong double graphs of the circumcoronene series of benzenoid Hm.


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## Body

## 1. Introduction and Preliminaries

For undetermined notations and terminologies, we refer the readers to read the book [1].LetƓV,E be a simple, finite connected graph, where the set of vertices is VƓ and the set of edges is EƓ. For every vertex x∈VƓ, the edge connecting x and z is denoted by xz. In graph Ɠ, the total number of edges that connects to each vertex is known as the degree of vertex. The number of connected vertices to a fixed vertex is known as neighborhood. The degree of a vertex is denoted by dx, where x∈VƓ. Hand-shaking lemma is very productive for calculating the size of a graph Ɠ.Lemma 1.
If a graphƓ is having size k, then(1)∑x∈VƓdegx=2k.In chemical graph theory, topological indices show a significant role in assisting chemists for modeling the molecular structure of chemical compounds and studying their chemical and physical characteristics. In chemistry, discovery of the drugs commonly relies on the topological descriptors. Drugs are characterized as molecular graphs, where graphs considered are simple with no multiple edges and no cycle formation. These topological descriptors provide information of a chemical compound based on the arrangement of its atoms and their bonds. A wide range of topological indices have been studied, and some of the more frequent forms of topological indices include degree-based, distance-based topological indices, and counting-related polynomials. In the topological indices, very famous and the oldest index is the Wiener indexWƓ.The Wiener index [2] is defined as follows:(2)WƓ=12∑x,zdx,z,where dx,z is the distance among vertices x and z of a graph Ɠ.A graphƓ’s geometric arithmetic index GA [3] is defined as follows:(3)GAƓ=∑xz∈EƓ2dxdzdx+dz.A graphƓ’s atomic bond connectivity index ABC [4] is defined as follows:(4)ABCƓ=∑xz∈EƓdx+dz−2dxdz.A graphƓ’s forgotten index F [5] is defined as follows:(5)FƓ=∑xz∈EƓdx2+dz2.A graphƓ’s inverse sum indeg index ISI [6] is defined as follows:(6)ISIƓ=∑xz∈EƓ11/dx+1/dz.A graphƓ’s general inverse sum indeg index ISIα,β [7] is defined as follows:(7)ISIα,βƓ=∑xz∈EƓdxdzαdx+dzβ,where α and β are the real numbers.A graphƓ’s first multiplicative-Zagreb PM1 and second multiplicative-Zagreb indices PM2 are defined [8] as follows:(8)PM1Ɠ=∏xz∈EƓdx2,(9)PM2Ɠ=∏xz∈EƓdx⋅dz.It is also possible to write the first multiplicative-Zagreb indexPM1 [9] for Ɠ as follows:(10)PM1Ɠ=∏xz∈EƓdx+dz.Imran et al. [10–12] studied the edge Mostar index of nanostructures and chemical structures by using graph operations and also computed the eccentric connectivity polynomial of connected graphs and Mostar indices for melem chain nanostructures. For more details about topological indices, we refer the works of Xiong et al. [13], Hong et al. [14], Alaeiyan et al. [15], Ch et al. [16], and Sardar et al. [17].Definition 1.
The well-known family of the benzenoid molecular graph is circumcoronene series of benzenoidHm, where m≥1 [18]. This family of graph constructed exclusively from benzene C6 on circumference. Certain main members of circumcoronene series of benzenoid are benzene H1, coronene H2, circumcoronene H3, and circumcircumcoronene H4 [19]. Generally, circumcoronene series of benzenoid Hm is shown in Figure 1.Figure 1
Circumcoronene series of benzenoidH1,H2,H3,andHm.Definition 2.
In order to make a double graphDHm of a graph G, take two copies of the graph G and join the nodes in each copy with their neighbors in the other copy [20]. For example, the graph H1 and its double graph DH1 are shown in Figure 2. In double graph of circumcoronene series of benzenoid, there are 12m2 vertices and 49m2−3m edges, respectively. In DHm, we have 12m vertices of degree 4 and 12m2−m vertices of degree.Figure 2
Circumcoronene series of benzenoidH1 and its double graph DH1.Definition 3.
Consider the two copies of graphG, and by joining the closed neighborhoods of one graph’s vertex to the vertex in an adjacent graph, one can obtain the strong double graph SDG of graph G [21]. For example, strong double graph of graph H1 is shown in Figure 3.
This study is laid out as follows. We will examine some vertex-based topological indices of double and strong double graphs of circumcoronene series of benzenoidHm in Sections 2 and 4, respectively. The comparison is given in Sections 3 and 5. In Section 6, we provide final remarks for the whole study.Figure 3
Circumcoronene series of benzenoidH1 and its strong double graph SDH1.


## 2. Degree-Based Topological Indices of Double Graph of Circumcoronene Series of Benzenoid GraphHm

This section contains a calculation of the degree-based indices of the double graph of circumcoronene series of benzenoidHm.Theorem 1.
LetDHm be the double graph of circumcoronene series of benzenoid graph Hm; then, the geometric arithmetic index of DHm is(11)GADHm=96m−9665+36m2−60m+48.Proof.
In the double graph of circumcoronene series of benzenoid, there are12m2 vertices and 49m2−3m edges, respectively. There are 12m vertices in DHm of degree 4 and 12m2−m of degree 6.
We separate the edges ofDHm into the edges of the typeEdx,dz, where xz is an edge. In DHm, we get edge of types E4,4 and E4,6 and E6,6. A list of their edges is given in Table 1.
By using Table1 and equation (1), the result that we obtain is(12)GAƓ=∑xz∈EG2dxdzdx+dz.GADHm=E4,4∑xz∈EDHm2dxdzdx+dz+E4,6∑xz∈EDHm2dxdzdx+dz+E6,6∑xz∈EDHm2dxdzdx+dz.GADHm=242168+48m−122410+36m2−60m+2423612.GADHm=24+48m−1245+36m2−60m+24.GADHm=96m−9665+36m2−60m+48.Table 1
Separation of edges.
Edx,dzE4,4E4,6E6,6Number of edges2448m−136m2−60m+24Theorem 2.
LetDHm be the double graph of circumcoronene series of the benzenoid graph Hm; then, the ABC index of DHm is(13)ABCDHm=63+6m2−10m+452+163m−1.Proof.
By using Table1 and equation (4), the result that we obtain is(14)ABCDHm=E4,4∑xz∈EDHmdx+dz−2dxdz+E4,6∑xz∈EDHmdx+dz−2dxdz+E6,6∑xz∈EDHmdx+dz−2dxdz.=244+4−244+48m−14+6−246+36m2−60m+246+6−266.=66+48m−113+36m2−60m+24518.ABCDHm=63+6m2−10m+452+163m−1.Theorem 3.
LetDHm be the double graph of circumcoronene series of benzenoid graph Hm; then, the forgotten index ofDHm is(15)FDHm=2592m2−1824m.Proof.
By using Table1 and equation (5), the result that we obtain is(16)FDHm=E4,4∑xz∈EDHmdx2+dz2+E4,6∑xz∈EDHmdx2+dz2+E6,6∑xz∈EDHmdx2+dz2.=2442+42+48m−142+62+36m2−60m+2462+62=768+2496m−1+36m2−60m+2472.FDHm=2592m2−1824m.Theorem 4.
LetDHm be the double graph of circumcoronene series of the benzenoid graph Hm; then, the inverse sum indeg index of DHm is(17)ISIDHm=108m2−3245m+245.Proof.
By using Table1 and equation (6), the result that we obtain is(18)ISIDHm=E4,4∑xz∈EDHmdxdzdx+dz+E4,6∑xz∈EDHmdxdzdx+dz+E6,6∑xz∈EDHmdxdzdx+dz=24444+4+48m−1464+6+36m2−60m+24666+6=48+48m−1125+36m2−60m+243,ISIDHm=108m2−3245m+245.Theorem 5.
LetDHm be the double graph of circumcoronene series of the benzenoid graph Hm; then, the general inverse sum indeg index (ISIα,β) of DHm is(19)ISIα,βDHm=4p16α8β+8p16pα41+pβ.Proof.
By using Table1 and equation (7), the result that we obtain is(20)ISIα,βDHm=E4,4∑xz∈EDHmdxdzαdx+dzβ+E4,6∑xz∈EDHmdxdzαdx+dzβ+E6,6∑xz∈EDHmdxdzαdx+dzβ=2444α4+4β+48m−146α4+6β+36m2−60m+2466α6+6β=2416α8β+48m−124α10β+36m2−60m+2436α12β,where α and β are the real numbers.Theorem 6.
LetDHM be the double graph of circumcoronene series of the benzenoid graph Hm; then, the first multiplicative-Zagreb index of DHm is(21)PM1DHm=3m−213271040m−12.Proof.
By using Table1 and equation (10), the result that we obtain is(22)PM1DHm=E4,4∏xz∈EDHmdx+dz×E4,6∏xz∈EDHmdx+dz×E6,6∏xz∈EDHmdx+dz,PM1DHm=248×48m−110×36m2−60m+2412,PM1DHm=192×480m−480×432m2−720m+288,PM1DHm=3m−213271040m−12.Theorem 7.
LetDHm be the double graph of circumcoronene series of the benzenoid graph Hm; then, the second multiplicative-Zagreb index of DHm is(23)PM2DHm=m−23573308928m−12.Proof.
By using Table1 and equation (9), the result that we obtain is(24)PM2DHm=E4,4∏xz∈EDHmdx⋅dz×E4,6∏xz∈EDHmdx⋅dz×E6,6∏xz∈EDHmdx⋅dz,PM2DHm=2416×48m−124×36m2−60m+2436,PM2DHm=442368m−1×1296m2−2160m+864,PM2DHm=m−23573308928m−12.


## 3. Comparison

In this section, we present a numerical and graphical comparison of topological indices that included the first multiplicative-Zagreb indexPM1, general inverse sum indeg index (ISIα,β), atom bond connectivity index (ABC), forgotten index (F), geometric arithmetic index GA, second multiplicative-Zagreb index PM2, and inverse sum indeg index ISI for m = 1, 2, 3, 4, …, 10 for the double graph of circumcoronene series of the benzenoid graph DHm, as given in Table 2 and Figure 4.Table 2
Computation of topological indices of double graph of circumcoronene series of benzenoidDHm.
mGADHmABCDHmFDHmISIDHmPM1DHmPM2DHm12414.69776848002119.0367.7106720307.205.3084×1077.6441×1083286.06158.6717856782.403.7159×1085.3509×1094525.09287.58341761473.61.1944×1091.7199×10105836.12454.42556802380.82.7604×1093.9749×101061219.2659.248236035045.3084×1097.6441×101071674.2901.981.1424×1054843.29.0774×1091.3071×101182201.21182.71.5129×1056398.41.4306×10102.0601×101192800.21501.31.9353×1058169.62.1234×10103.0576×1011103471.31857.92.4096×10510156.83.0099×10104.3342×1011Figure 4
Graphical representation of topological indices of double graph of circumcoronene series of benzenoidHm.


## 4. Degree-Based Topological Indices of Strong Double Graphs of Circumcoronene Series of Benzenoid GraphHm

This section contains a calculation of the degree-based indices of the strong double graph of circumcoronene series of benzenoidHm. Figure 3 shows the strong double graph of H1.Theorem 8.
LetSDHM be the double graph of circumcoronene series of the benzenoid graph Hm; then, the geometric arithmetic index of SDHm is(25)GASDHm=8m−835+42m2−60m+48.Proof.
In the strong double graph of circumcoronene series of benzenoid, there are12m2 vertices and 67m2−2m edges, respectively. There are 12m vertices in SDHm of degree 5 and 12mm2−1 of degree 7.
We separate the edges ofSDHm into the edges of the typeEdx,dz, where xz is an edge. In SDHm, we get edge of types E5,5 and E5,7 and E7,7. A list of their edges is given in Table 3.
By using Table3 and equation (1), the result that we obtain is(26)GAƓ=∑xz∈EG2dxdzdx+dz,GASDHm=E5,5∑xz∈ESDHm2dxdzdx+dz+E5,7∑xz∈ESDHm2dxdzdx+dz+E7,7∑xz∈ESDHm2dxdzdx+dz,GASDHm=6m+2422510+48m−123512+42m2−66m+2424914,GASDHm=6m+24+48m−1356+42m2−66m+24,GASDHm=8m−835+42m2−60m+48.Table 3
Separation of edges.
Edx,dzE5,5E5,7E7,7Number of edges6m+2448m−142m2−66m+24Theorem 9.
LetSDHm be the strong double graph of circumcoronene series of the benzenoid graph Hm; then, the ABC index of SDHm is (27)ABCSDHm=240m−2407+84m+336235+12m−4873m−1.Proof.
By using Table3 and equation (4), the result that we obtain is(28)ABCSDHm=E5,5∑xz∈ESDHmdx+dz−2dxdz+E5,7∑xz∈ESDHmdx+dz−2dxdz+E7,7∑xz∈ESDHmdx+dz−2dxdz=6m+245+5−255+48m−15+7−257+42m2−66m+247+7−277=6m+2485+48m−127+42m2−66m+24127,ABCSDHm=240m−2407+84m+336235+12m−4873m−1.Theorem 10.
LetSDHm be the strong double graph of circumcoronene series of the benzenoid graph Hm; then, the forgotten index of SDHm is(29)FSDHm=4116m2−2616m.Proof.
By using Table3 and equation (5), the result that we obtain is(30)FSDHm=E5,5∑xz∈ESDHmdx2+dz2+E5,7∑xz∈ESDHmdx2+dz2+E7,7∑xz∈ESDHmdx2+dz2=6m+2452+52+48m−152+72+42m2−66m+2472+72=300m+1200+3552m−1+42m2−66m+2498,FSDHm=4116m2−2616m.Theorem 11.
LetSDHm be the strong double graph of circumcoronene series of the benzenoid graph Hm; then, the inverse sum indeg index of SDHm is(31)ISISDHm=147m2−76m+4.Proof.
By using Table3 and equation (6), the result that we obtain is(32)ISISDHm=E5,5∑xz∈ESDHmdxdzdx+dz+E5,7∑xz∈ESDHmdxdzdx+dz+E7,7∑xz∈ESDHmdxdzdx+dz=6m+24555+5+48m−1575+7+42m2−66m+24777+7=15m+60+140m−1+42m2−66m+244914,ISISDHm=147m2−76m+4.Theorem 12.
LetSDHm be the strong double graph of circumcoronene series of the benzenoid graph Hm; then, the general inverse sum indeg index (ISIα,β) of SDHm is(33)ISIα,βSDHm=6m+2425α10β+48m−135α12β+42m2−66m+2449α14β.Proof.
By using Table3 and equation (7), the result that we obtain is(34)ISIα,βSDHm=E5,5∑xz∈ESDHmdxdzαdx+dzβ+E5,7∑xz∈ESDHmdxdzαdx+dzβ+E7,7∑xz∈ESDHmdxdzαdx+dzβ=6m+2455α5+5β+48m−157α5+7β+42m2−66m+2477α7+7β=6m+2425α10β+48m−135α12β+42m2−66m+2449α14β,where α and β are the real numbers.Theorem 13.
LetSDHm be the strong double graph of circumcoronene series of the benzenoid graph Hm; then, the first multiplicative-Zagreb index of SDHm is(35)PM1SDHm=20321280m−47m+4m−12.Proof.
By using Table3 and equation (10), the result that we obtain is(36)PM1SDHm=E5,5∑xz∈ESDHmdx+dz×E5,7∑xz∈ESDHmdx+dz×E7,7∑xz∈ESDHmdx+dz,PM1SDHm=6m+2410×48m−112×42m2−66m+2414,PM1SDHm=60m+240×576m−576×588m2−924m+336,PM1SDHm=20321280m−47m+4m−12.Theorem 14.
LetSDHm be the strong double graph of circumcoronene series of the benzenoid graph Hm; then, the second multiplicative-Zagreb index of SDHm is(37)PM2SDHm=518616000m−47m+4m−12.Proof.
By using Table3 and equation (9), the result that we obtain is(38)PM2SDHm=E5,5∏xz∈ESDHmdx⋅dz×E5,7∏xz∈ESDHmdx⋅dz×E7,7∏xz∈ESDHmdx⋅dz,PM2SDHm=6m+2425×48m−135×42m2−66m+2449,PM2SDHm=252000m+3m−1×2058m2−3234m+1176,PM2SDHm=518616000m−47m+4m−12.


## 5. Comparison

In this section, we present a numerical and graphical comparison of topological indices that included the first multiplicative-Zagreb indexPM1, general inverse sum indeg index (ISIα,β), atom bond connectivity index (ABC), forgotten index (F), geometric arithmetic index GA, second multiplicative-Zagreb index PM2, and inverse sum indeg index ISI for m = 1, 2, 3, 4, …, 10 for the strong double graph of circumcoronene series of the benzenoid graph SDHm, as given in Table 4 and Figure 5.Table 4
Computation of topological indices of strong double graph of circumcoronene series of benzenoidSDHm.
mGASDHmABCSDHmFSDHmISISDHmPM1SDHmPM2SDHm13016.970150075002143.3375.715112324401.7418 × 1084.4453 × 1093340.66176.032919610991.3818 × 1093.5266 × 10104621.99317.915539220525.0165 × 1091.2802 × 10115987.32501.378982032991.2959 × 10103.3073 × 101161436.6726.3913248048402.7579 × 10107.0384 × 101179070.0993.0018337266755.1732 × 10101.3202 × 101282587.31301.124249688048.8763 × 10102.2653 × 101293288.61650.9309852112271.4250 × 10113.6368 × 1012104074.02042.2385440139442.1728 × 10115.5450 × 1012Figure 5
Graphical representation of topological indices of the strong double graph of circumcoronene series of benzenoidHm.


## 6. Conclusion

We have computed the closed formulae of topological indices such as the first multiplicative-Zagreb indexPM1, general inverse sum indeg index (ISIα,β), atom bond connectivity index (ABC), forgotten index (F), geometric arithmetic index GA, second multiplicative-Zagreb index PM2, and inverse sum indeg index ISI of double and strong double graphs of circumcoronene series of benzenoid Hmm≥1. Chemical compounds can be studied by these indices in order to understand their diverse properties. The geometric structure and comparison of obtained results are shown graphically and numerically. Those results are convenient for further study as they do not include any polynomial.



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