# Topological Indices of Derived Networks of Benzene Ring Embedded inP-Type Surface on 2D

Authors: Feng Yin; Muhammad Numan; Saad Ihsan Butt; Adnan Aslam; Andleeb Kausar
Journal: Journal of Chemistry
 (2021)

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## Abstract

Topological index (TI) is a numerical number assigned to the molecular structure that is used for correlation analysis in pharmacology, toxicology, and theoretical and environmental chemistry. Benzene ring embedded in theP-type surface on 2D network has stability similar to C60 and can be defined as 3D linkage of C8 rings. This structure is the simplest possible tilling of the periodic minimal surface P which contains one type of carbon atom. In this paper, we compute general Randić, general Zagreb, general sum-connectivity, first Zagreb, second Zagreb, and ABC and GA indices of two operations (simple medial and stellation) of 2D network of benzene ring. Also, the exact expressions of ABC4 and GA5 indices of these structures are computed.


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## Body

## 1. Introduction and Preliminaries

All the graphs in this work are finite and connected. Letℋ be a graph with vertex set and edge set denoted by Vℋ and Eℋ, respectively. We denote the degree of a vertex u∈VG by du and it is the number of edges incident to u. The neighbor of a vertex v is a vertex u such that uv∈EG. The neighborhood of a vertex u is the set of all its neighbors and is denoted by Nu. Let Su be the sum of degrees of all the vertices that are adjacent to u. In other words,(1)Su=∑v∈Nudv,whereNu=v∈Vℋ:uv∈Eℋ.For more insight on basic definitions and terminologies of graph theory, see [1].In this paper, we consider two operations, stellation and simple medial of 2D network of benzene ring. The medial of a graphℋ, denoted by Mℋ, is defined as follows: we put a new vertex in the middle of every old edge of ℋ and the new vertices have an edge if they lie on the consecutive edges. Note that the medial of a graph ℋ is a 4-regular planner graph and not necessarily simple. Sjostrand [2] introduced the idea of transforming the graph with multiple edges and loops in to a simple graph by finite sequence of double edge swaps. If Mℋ is not simple, we transform the graph into simple graph and call it the simple medial of ℋ, denoted by SMℋ. Stellation of a graph planar ℋ, denoted as Stℋ, is obtained by putting a vertex in every face of ℋ and then we join this vertex to each vertex of respective face.In the last couple of decades, topological and graph theoretical models have shown applications in many scientific research areas such as theoretical physics, chemistry, pharmaceutical chemistry, and toxicology. The interaction of graph theory with chemistry has enriched both the field. Topological index/descriptor is a numerical number attached to a molecular graph which is expected to predict certain physical or chemical properties of the underlying molecular structure. The simplest topological descriptors one can attach to a graphℋ is its order and size. The importance of the topological indices is because of their use in quantitative structure activity relationship (QSAR)/quantitative structure property relationship (QSPR). The first topological index was introduced by Weiner in 1947, who showed that the index is well correlated with boiling point of alkanes. In 1975, the first degree based topological index was proposed by Milan Randić [3]. After that many degree-based topological indices were defined which were found to be useful in modeling the properties of organic molecules. Few of the important degree-based topological indices are presented in Table 1.Table 1
Degree-based topological descriptors.
Topological descriptorsMathematical formsGeneral Randić index [4,5]Rαℋ=∑uv∈EℋdudvαGeneral sum-connectivity index [6]χαℋ=∑uv∈Eℋdu+dvαFirst general Zagreb index [7]Mαℋ=∑u∈VℋduαRandić index [3]R−1/2ℋ=∑uv∈Eℋ1/dudvFirst Zagreb index [8]M1ℋ=∑uv∈Eℋdu+dvSecond Zagreb index [8]M2ℋ=∑uv∈Eℋdu×dvFirst multiplicative Zagreb index [9]PM1ℋ=∏uv∈Eℋdu+dvSecond multiplicative Zagreb index [10]PM2ℋ=∏uv∈EℋdudvHyper-Zagreb index [11]HMℋ=∑uv∈Eℋdu+dv2.Atom-bond connectivity index [12]ABCℋ=∑uv∈Eℋdu+dv−2/dudvGeometric arithmetic index [13]GAℋ=∑uv∈Eℋ2dudv/du+dvFourth version of atom-bond connectivity index [14]ABC4ℋ=∑uv∈EℋSu+Sv−2/SuSvFifth version of geometric arithmetic index [15]GA5ℋ=∑uv∈Eℋ2SuSv/Su+SvThe Randić index was first named as branching index and is found appropriate for calculating the extent of branching of the carbon atom skeleton of saturated hydrocarbons. The first and second Zagreb indices were first introduced by Gutman and Transjistic in [8] and applied to branching problem. The Zagreb indices and their different variants are used to study chirality [16], molecular complexity [17, 18], ZE isomerism [19], and heterosystems [20]. The overall Zagreb indices are used to derive multilinear regression models. The importance of ABC index is due to its correlation with the thermodynamic properties of alkanes, see [21, 22]. Details on the computation of topological indices of graphs can be seen in [23–25].


## 2. Topological Indices of Simple Medial ofPm,n

The preparation [26] of C60 leads to assumption about the stability of other crystalline forms of three coordinated carbons. In particular, Mackay and Terrones [27] raised the interesting prospect of creating possible tricoordinated solid carbon forms by lining the infinite periodic minimal surfaces known as P and D. These surfaces divide the space into two unconnected labyrinths. OKeeffe et al. [28] reported the results of initial calculations of molecular dynamic relaxation in the simplest treatment, which contains only one type of carbon atom. These structures contain six- and eight-membered rings in ratio of 2:3 and their primitive single cells have only 24 atoms. The stability of this structure is similar to C60 and it can be defined as a three-dimensional connection of C8 rings. This structure is the simplest possible treatment of the periodic minimum surface P, which has only one type of carbon atom. From now onward, we denote the molecular structure of 2D network of benzene ring embedded in P-type surface by Pm,n. Figure 1 depicts the molecular graph of Pm,n.Figure 1
Graph ofPm,n.Note thatPm,n contains 24mn vertices and 32mn−2m−2n edges. The medial of Pm,n is obtained as follows: we put a new vertex in the middle of every old edge of Pm,n and the new vertices have an edge if they lie on the consecutive edges. The graph of medial of Pm,n is depicted in Figure 2. Observe that the graph of medial of Pm,n contains multiple edges. It can be made simple by using the double edge swaps defined by Sjostrand [2]. Figure 3 depicts the graph of simple medial of Pm,n and we denote it by SMPm,n. By a simple calculation, we can compute that SMPm,n contains 32mn−2m−2n vertices and 64mn−20m−20n+12 edges. Suppose Vi=u∈VSMPm,n:du=i and Ei,j=uv∈ESMPm,n:du=i,dv=j. Let ni and ei,j be the cardinalities of Vi and Ei,j, respectively.Figure 2
Molecular graph of medial ofPm,n.Figure 3
Molecular graphSMPm,n of simple medial of Pm,n.Theorem 1.
LetK be the graph of SMPm,n and α is a real number, then we have(1)
MαK=16m+16n−122α+32mn−18m−18n+124α,(2)
RαK=4m+4n22α+24m+24n−2423α+64mn−48m−48n+3624α,(3)
χαK=4m+4n22α+24m+24n−246α+64mn−48m−48n+3623α,(4)
ABCK=1/228m+28n−24+1/23/264mn−48m−48n+36,(5)
GAK=4m+4n+2/324m+24n−24+64mn−48m−48n+36,(6)
PM1K=2192mn−136m−136n+108×624m+24n−24, and(7)
PM2K=2256mn−112m−112n+72.Proof.
We can partitionVK into three sets based on vertex degrees. Table 2 shows this partition. By using the values presented in Table 2, the general Zagreb index of K can be computed as follows:(2)MαK=∑u∈Vℋduα=16m+16n−122α+32mn−18m−18n+124α.
Similarly, we can partitionEK into three sets based on the degree of end vertices of each edge. Table 3 shows this partition. By using the values presented in Table 3, the values of Rα,χα,ABC,GA,PM1, and PM2 indices of K can be computed as(3)RαK=∑uv∈EKdudvα=4m+4n2×2α+24m+24n−242×4α+64mn−48m−48n+324×4α=4m+4n22α+24m+24n−2423α+64mn−48m−48n+3624α,χαK=∑uv∈EKdu+dvα=4m+4n2+2α+24m+24n−242+4α+64mn−48m−48n+324+4α=4m+4n22α+24m+24n−246α+64mn−48m−48n+3623α,ABCK=∑uv∈EKdu+dv−2dudv=4m+4n2+2−22×2+24m+24n−242+4−22×4+64mn−48m−48n+324+4−24×4=1228m+28n−24+123264mn−48m−48n+36,GAK=∑uv∈EK2dudvdu+dv=4m+4n22×22+2+24m+24n−2422×42+4+64mn−48m−48n+3224×44+4=4m+4n+2324m+24n−24+64mn−48m−48n+36,PM1K=∏uv∈EKdu+dv=2+24m+4n×2+424m+24n−24×4+464mn−48m−48n+32=2192mn−136m−136n+108×624m+24n−24,PM2K=∏uv∈EKdudv=2×24m+4n×2×424m+24n−24×4×464mn−48m−48n+32=2256mn−112m−112n+72.Table 2
Vertex partition ofK.
Vi24ni16m+16n−1232mn−18m−18n+12Table 3
Edge partitionEi,j of K.
Ei,jE2,2E2,4E4,4ei,j4m + 4n24m + 24n − 2464mn − 48m − 48n + 36From Theorem1, we can compute the values of Randić, first Zagreb, second Zagreb, and hyper-Zagreb index of K.Corollary 1.
LetK be the graph of simple medial of Pm,n, then we have(1)
R−1/2K=4m+4n/2+24m+24n−24/22+64mn−48m−48n+36/4,(2)
M1K=512mn−224m+n+144,(3)
M2K=1024mn−560m+n+384, and(4)
HMK=4096mn−2144m+n+1440.Next, we will compute theABC4 and GA5 indices of K. For this, we need to find the edge partition Si,j of the graph K, where Si,j=uv∈EK:Su=i,Sv=j. Let mi,j denote the cardinality of the set Si,j. The edge partition Si,j of K is given in Table 4.Table 4
Edge partitionSi,j of K.
Si,jS6,6S6,10S8,10S8,12S8,14S10,10mi,j4m + 4n8m + 8n4m + 4n4m + 4n − 84m + 4n − 84Si,jS10,14S12,14S12,16S14,16S16,16mi,j4m + 4n − 84m + 4n − 84m + 4n − 84m + 4n − 864mn − 60m − 60n + 56Theorem 2.
LetK be the graph of simple medial of Pm,n, then we have(4)1ABC4K=13524m+4n+7308m+8n+154m+4n+12574m+4n−8+12512+174m+4n−8+1213214m+4n−8+12124m+4n−8+11704m+4n−8+1815264mn−60m−60n+56,2GA5K=4m+4n+14158m+8n+4954m+4n+41174m+4n−8+4+13354m+4n−8+213424m+4n−8+4734m+4n−8+415144m+4n−8+64mn−60m−60n+56.Proof.
The edge partition ofK depending on the sum of degree of end vertices is presented in Table 4. The result follows by using the values from Table 4 in the definition of ABC4K and GA5K.


## 3. Topological Indices of Stellation ofPm,n

Letℒ be the molecular graph of stellation of Pm,n. It is obtained adding a vertex in each face of Pm,n and then joining this vertex to each vertex of the respective face. The graph of ℒ is shown in Figure 4. In ℒ, there are 32mn−2n+1 vertices and 96mn−22m−22n+12 edges. Suppose Vi=u∈Vℒ:du=i and Ei,j=uv∈Eℒ:du=i,dv=j. Let ni and ei,j be the cardinalities of the vertex set Vi and edge set Ei,j, respectively.Figure 4
Molecular graph ofstPm,n.Theorem 3.
Letℒ be the graph of stellation of Pm,n and α is a real number, then we have(1)
Mαℒ=8m+8n−43α+8mn−4m−4n+44α+8m+8n−85α+20mn−12m−12n+86α+2mn−6m−6n8α+2mn−m−n+112α,(2)
Rαℒ=4m+4n32α+8m+8n−815α+8m+8n−418α+4m+4n20α+24mn−16m−16n+1224α+24mn−16m−16n+1248α+4m+4n−452α+12m+12n−1630α+4m+4n−840α+4m+4n60α+32mn−26m−26n+2062α+16mn−12m−12n+872α,(3)
χαℒ=4m+4n6α+8m+8n−88α+12m+12n−432α+24mn−16m−16n+1210α+8mn−4m−4n+416α+4m+4n−410α+12m+12n−1611α+4m+4n−813α+4m+4n17α+32mn−26m−26n+2012α+16mn−12m−12n+814α+16mn−12m−12n+818α,(4)
ABCℒ=243+242+1610+162/3+8mn+414−163−42−1310−4/3+810+235+630+110−122/5m+n+−214+123+1010+42+82/3+−82−830−2110−810/5+4,(5)
GAℒ=32+486/5+322/3+923/7mn+165/9−326/5+2430/11+1610/13−82/3+5015/17−623/7m+n+246/5−3230/11−3210/13+82/3+463/7−215,(6)
PM1ℒ=2152mn−76m−76n+56×364mn−22m−22n+28×524mn−12m−12n+8×716mn−12m−12n+8×1112m+12n−16×134m+4n−8×174m+4n, and(7)
PM2ℒ=2216mn−100m−100n+64×3144mn−60m−60n+48×540m+40n−40.Proof.
We can partitionVℒ into six sets based on vertex degrees. Table 5 shows this partition. By using the values presented in Table 5, the general Zagreb index of ℒ can be computed as(5)Mαℒ=∑u∈Vℒduα=8m+8n−43α+8mn−4m−4n+44α+8m+8n−45α+20mn−12m−12n+86α+2mn−m−n8α+2mn−m−n+112α.
Similarly, we can partitionEℒ into three sets based on the degree of end vertices of each edge. Table 6 shows this partition. By using the values presented in Table 6, the values of Rα,χα,ABC,GA,PM1, and PM2 indices of ℒ can be computed as(6)Rαℒ=∑uv∈Eℒdudvα=4m+4n3×3α+24mn−16m−16n+123×5α+8m+8n−83×6α+4m+4n4×5α+8m+8n−84×6α+8mn−4m−4n+44×12α+4m+4n−45×5α+12m+12n−165×6α+4m+4n−85×8α+4m+4n5×12α+32mn−26m−26n+206×6α+16mn−12m−12n+86×8α+16mn−12m−12n+86×12α=4m+4n32α+8m+8n−815α+8m+8n−418α+4m+4n20α+24mn−16m−16n+1224α+24mn−16m−16n+1248α+4m+4n−452α+12m+12n−1630α+4m+4n−840α+4m+4n60α+32mn−26m−26n+2062α+16mn−12m−12n+872α.χαℒ=∑uv∈Eℒdu+dvα=4m+4n3+3α24mn−16m−16n+123+5α+8m+8n−83+6α+4m+4n4+5α+8m+8n−84+6α+8mn−4m−4n+44+12α+4m+4n−45+5α+12m+12n−165+6α+4m+4n−85+8α+4m+4n5+12α+32mn−26m−26n+206+6α+16mn−12m−12n+86+8α+16mn−12m−12n+86+12α=4m+4n6α+8m+8n−88α+12m+12n−432α+24mn−16m−16n+1210α+8mn−4m−4n+416α+4m+4n−410α+12m+12n−1611α+4m+4n−813α+4m+4n17α+32mn−26m−26n+2012α+16mn−12m−12n+814α+16mn−12m−12n+818α.ABCℒ=∑uv∈Eℒdu+dv−2dudv=4m+4n3+3−23×3+24mn−16m−16n+123+5−23×5+8m+8n−83+6−23×6+4m+4n4+5−24×5+8m+8n−84+6−24×6+8mn−4m−4n+44+12−24×12+4m+4n−45+5−25×5+12m+12n−165+6−25×6+4m+4n−85+8−25×8+4m+4n5+12−25×12+32mn−26m−26n+206+6−26×6+16mn−12m−12n+86+8−26×8+16mn−12m−12n+86+12−26×12=243+242+1610+1623+8mn+414−163−42−1310−43+810+235+630+110−1225m+n+−214+123+1010+42+823+−82−830−2110−8105+4GAℒ=∑uv∈Eℒ2dudvdu+dv=4m+4n23×33+3+24mn−16m−16n+1223×53+5+8m+8n−823×63+6+4m+4n24×54+5+8m+8n−824×64+6+8mn−4m−4n+424×124+12+4m+4n−425×55+5+12m+12n−1625×65+6+4m+4n−825×85+8+4m+4n25×125+12+32mn−26m−26n+2026×66+6+16mn−12m−12n+826×86+8+16mn−12m−12n+826×126+12=32+4865+3223+9237mn+1659−3265+243011+161013−823+501517−6237m+n+2465−323011−321013+823+4637−215,PM1ℒ=∏uv∈Eℒdu+dv=3+34m+4n×3+524mn−16m−16n+12×3+68m+8n−8×4+54m+4n×4+68m+8n−8×4+128mn−4m−4n+4×5+54m+4n−4×5+612m+12n−16×5+84m+4n−8×5+124m+4n×6+632mn−26m−26n+20×6+816mn−12m−12n+8×6+1216mn−12m−12n+8=2152mn−76m−76n+56×364mn−22m−22n+28×524mn−12m−12n+8×716mn−12m−12n+8×1112m+12n−16×134m+4n−8×174m+4n,PM2ℒ=∏uv∈Eℒdudv=3×34m+4n×3×524mn−16m−16n+12×3×68m+8n−8×4×54m+4n×4×68m+8n−8×4×128mn−4m−4n+4×5×54m+4n−4×5×612m+12n−16×5×84m+4n−8×5×124m+4n×6×632mn−26m−26n+20×6×816mn−12m−12n+8×6×1216mn−12m−12n+8=2216mn−100m−100n+64×3144mn−60m−60n+48×540m+40n−40.Table 5
Vertex partition ofℒ.
Vi345ni8m+8n−48mn−4m−4n+48m+8n−8Vi6812ni20mn−12m−12n+82mn−m−n2mn−m−n+1Table 6
Edge partition ofEi,j of ℒ.
Ei,jE3,3E3,5E3,6ei,j4m+4n8m+8n−88m+8n−4Ei,jE4,5E4,6E4,12ei,j4m+4n24mn−16m−16n+128mn−4m−4n+4Ei,jE5,5E5,6E5,8ei,j4m+4n−412m+12n−164m+4n−8Ei,jE5,12E6,6E6,8ei,j4m+4n32mn−26m−26n+2016mn−12m−12n+8Ei,jE6,12ei,j16mn−12m−12n+8From Theorem3, we can compute the values of Randić, first Zagreb, second Zagreb, and hyper-Zagreb index of ℒ.Corollary 2.
Letℒ be the graph of stellation of Pm,n, then we have(1)
Rαℒ=24+24/48+16/72+16/3mn+8/15−8/18+4/20−16/24−16/48+12/30+4/40−4/60−12/72−11/5m+n−8/15−4/18−12/24+12/48−16/30−8/40+8/72+12/5,(2)
M1ℒ=1372mn−432m+n+260,(3)
M2ℒ=4032mn−1712m+n+1068, and(4)
HMℒ=17376mn−7296m+n+4740.Next, we will compute theABC4 and GA5 indices of ℒ. For this, we need to find the edge partition Si,j of the graph ℒ, where Si,j=uv∈Eℒ:Su=i,Sv=j. Let mi,j denote the cardinality of the set Si,j. The edge partition Si,j of ℒ is given in Table 7.Table 7
Edge partitionSi,j of ℒ.
Si,jS12,14S12,23S14,14S14,23S14,26mi,j844m + 4n − 888m + 8n − 16Si,jS14,28S14,30S23,28S23,30S26,28mi,j4m + 4n − 84m + 4n484m + 4n − 8Si,jS26,29S26,30S26,41S28,28S28,30mi,j2m + 2n − 44m + 4n − 84m + 4n − 82m + 2n − 48Si,jS28,41S28,46S28,60S29,30S29,41mi,j4m + 4n − 84m + 4n − 844m + 4n − 84m + 4n − 8Si,jS29,60S29,62S30,30S30,32S30,42mi,j84m + 4n − 162m + 2n8mn − 8m − 8n + 816mn − 16m − 16n + 16Si,jS30,60S30,62S30,64S32,42S41,42mi,j208m + 8n − 328mn − 12m − 12n + 2016mn − 16m − 16n + 164m + 4n − 8Si,jS41,46S41,60S41,62S42,42S42,46mi,j4m + 4n − 884m + 4n − 1616mn − 18m − 18n + 208m + 8n − 16Si,jS42,48S42,60S42,62S42,64mi,j16mn − 24m − 24n + 3284m + 4n − 1616mn − 24m − 24n + 40Theorem 4.
Letℒ be the graph of stellation of Pm,n, then we have(7)1ABC4ℒ=88221+823+4427+427321+69015+87721+2mn+2105+5815+3614+4345891+241538249+457+2267+2147+239962377+25510145+4808521189+219229287+613065+241041+2160310943+12161161+61722287−3827−823−4427−22737−69010−4777+2160022899+22567421271+418631+47378217−2m+n+−367−8345891−841538483−857−42674147−439962377−45510145−8808521189−419229287−1213065−441041−4160310943−24161161−121722287+108221+823+4427+1027321+6906+167721−8160022899−82567421271−1618631−167378217+877+423023+81515+455+46765205+47021+225323+216123+9030105+201115+2,2GA5ℒ=8353+1282137+641531+1284253+643047+641415+16mn+410511+4915+448311+823+818227+875455+887059+4118935+1628769+21957+8106667+8188687+832237+8172283−8353−1282137−641531−1924253−963047−961415+8179891+82542103+846523+865126−10m+n+−8915−848311−1618227−1675455−1687059−8118935−3228769−41957−16106667−16188687−1632237−16172283+8353+1282137+641531+3204253+1603047+1281415−32179891−322542103−3246523−1665113+164213+1632237+1669053+1621029+3243589+32615101+167017+166935+6116151+410511+82+16.4345891+241538249+457+2267+2147+239962377+25510145+4808521189+219229287+613065+241041+2160310943+12161161+61722287−3827−823−4427−22737−69010−4777+2160022899+22567421271+418631+47378217−2m+n+−367−8345891−841538483−857−4267−4147−439962377−45510145−8808521189−419229287−1213065−441041−4160310943−24161161−121722287+108221+823+4427+1027321+6906+167721−8160022899−82567421271−1618631−167378217+877+423023+81515+455+46765205+47021+225323+216123+9030105+201115+2.Proof.
The edge partition ofℒ depending on the sum of degree of end vertices is presented in Table 7. The result follows by using the values from Table 7 in the definition of ABC4ℒ and GA5ℒ.


## 4. Conclusion

In this work, we have considered two transformations (medial and stellation) on benzene ring embedded inP-type surface on 2D network. We have computed general Randić, general Zagreb, general sum-connectivity, first Zagreb, second Zagreb, first multiple Zagreb, second multiple Zagreb, ABC, GA, ABC4, and GA5 indices of these transformation graphs.



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