# On Analysis of Topological Aspects for Subdivision of Kragujevac Tree Networks

Authors: Salma Kanwal; Shanshan Shang; Muhammad Kamran Siddiqui; Tahira Sumbal Shaikh; Ammara Afzal; Anton Asare-Tuah
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, we have taken review of certain topological topological characteristics of subdivision and the line graph of subdivision of Kragujevac tree. A Kragujevac tree is denoted byK, K∈Kgq=r2t+1+1,r, with order r2t+1+1 and size r2t+1, respectively. We have computed the Zagreb polynomials, forgotten polynomial, and M-polynomial for Kragujevac tree. Moreover, we have computed topological indices like Zagreb-type indices, reduced reciprocal Randić indices, family of Gourava indices as well as forgotten index. Further, some topological indices that can be directly derived from M-polynomial, i.e., first and second Zagreb index, modified second Zagreb index, Randić and reciprocal Randić index, symmetric division and harmonic index, and inverse sum and augmented Zagreb index are also computed.


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## Body

## 1. Introduction

A topological index is a mathematical formula which is applicable to any graph that represents a molecular structure. It is also known as molecular descriptor. Using these indices, we can analyze the numerical out comes, and various physio-chemical characteristics of a molecule can be investigated. Therefore, use of topological indices is efficient way to avoid highly expensive and time taking laboratory experiments [1, 2].Recent innovation and advancement in mathematical chemistry and simulation of chemical graphs is due to the numerous research studies done in the field of chemical graph theory. Chemical graph theory enables the researchers to create a relationship between graph theory and chemical structures of compounds [3]. Due to the rapid increments in production of new drugs and chemical compounds by chemical industries, the workload in chemical graph theory has been increased. Molecular descriptors are playing significant role in chemistry, pharmacology, and biological and physical sciences [4]. Topological indexes via M-polynomials are the prominent component for studying the various properties of chemical graphs [5, 6].Topological indices do not relay on labeling and graphical representation of some graphs. Degrees and distances between vertices give rise to some chemical invariants called topological indices. Gutman and Trinajstić [7] three decades before put forward the Zagreb indices. Later on, Balaban et al. [8, 9] termed the Zagreb indices as Zagreb group indices.Then, the first Zagreb index is expressed as follows:(1)M1H=∑p∈VHdp2=∑kp∈EHdk+dp.The second Zagreb index is expressed as follows:(2)M2H=∑kp∈EHdkdp.Fath-Tabar [10] presented the third Zagreb index in 2011 symbolised by M3H for some graph H and is expressed as follows:(3)M3H=∑kp∈EHdk−dp.Zagreb polynomials were brought into notice in the same year by the same author, named as1st, 2nd, and 3rd Zagreb polynomials and are presented as follows:(4)M1H,w=∑kp∈EHwdk+dp,M2H,w=∑kp∈EHwdkdp,M3H,w=∑kp∈EHwdk−dp,respectively. Another variant of Zagreb indices is modified Zagreb indices [11] which also deal with degrees of vertices. The first and second modified Zagreb indices are denoted by M1mH and M2mH individually and are as follows:(5)M1mH=∑kp∈EH1dk+dp,M2mH=∑kp∈EH1dkdp.The hyper-Zagreb index was presented by Shirdel et al. [12] as follows:(6)HMH=∑kp∈EHdk+dp2.The redefined Zagreb indices were put forward in 2013 by RanHini et al. [13] and were defined as follows:(7)redefined first Zagreb Index=ReM1H=∑kp∈EHdk+dpdkdp,(8)redefined second Zagreb Index=ReM2H=∑kp∈EHdkdpdk+dp,(9)redefined third Zagreb Index=ReM3H=∑kp∈EHdkdpdk+dp.The forgotten index symbolised byFH for a graph H was introduced in [14] by Furtula et al. and defined as follows:(10)FH=∑p∈VHdp3=∑kp∈EHdk2+dp2.Parallel to the notion of the forgotten topological index is that of forgotten polynomial. Forgotten polynomial of a graphH is given by(11)FH,w=∑kp∈EHwdk2+dp2.Reciprocal Randić, reduced second Zagreb, and reduced reciprocal Randić indices were proposed by I. Gutman et al. [15] in 2014 and are defined as follows:(12)RRH=∑kp∈EHdkdp,(13)RM2H=∑kp∈EHdk−1dp−1,(14)RRRH=∑kp∈EHdk−1dp−1.Many new graph invariants [16–19] which are well-known family of Gourava indices were brought into existence by V. R. Kulli in 2017 and are described as follows:(15)first Gourava index=G1OH=∑kp∈EHdk+dp+dkdp,(16)second Gourava index=G2OH=∑kp∈EHdk+dpdkdp,(17)product connectivity Gourava index=PGOH=∑kp∈EH1dk+dpdkdp,(18)sum connectivity Gourava index=SGOH=∑kp∈EH1dk+dp+dkdp,(19)first hyper−Gourava index=HGO1H=∑kp∈EHdk+dp+dkdp2,(20)second hyper Gourava index=HGO2H=∑kp∈EHdk+dpdkdp2.M-polynomial [20] is given as(21)MH;w,x=∑δ≤r≤s≤ΔyrsHwrxs.Withδ and Δ as, respectively, minimum and maximum of vertex degrees in graph, H and yrs represent the number of edges kp∈EH such that dk,dp=r,s (see [21]). We can deduce various indices from M-polynomial such as follows:(22)1stZagreb index=M1H=Dw+DxMH;w,xw=x=1,(23)2ndZagreb index=M2H=DwDxMH;w,xw=x=1,(24)modified2ndZagreb index=M2mH=SwSxMH;w,xw=x=1,(25)Randic`index=RαH=SwαSxαMH;w,xw=x=1=∑kp∈EHdkdpα,(26)inverse Randic`index=RRH=DwαDxαMH;w,xw=x=1=∑kp∈EHdkdp.The symmetric division deg index is as follows:(27)SDDH=DwSx+DxSwMH;w,xw=x=1=∑kp∈EHmindk,dpmaxdk,dp+maxdk,dpmindk,dp,(28)harmonic index=HH=2SwHMH;w,xw=x=1=∑kp∈EH2dk+dp,(29)inverse sum indeg index=ISIH=SwHDwDxMH;w,xw=x=1=∑kp∈EHdkdpdk+dp,and the augmented Zagreb index is as follows:(30)AH=Sw3Q−2Dw3Dx3MH;w,xw=x=1=∑kp∈EHdkdpdk+dp−23,where(31)DwMH;w,x=w∂MH;w,x∂w,DxMH;w,x=x∂MH;w,x∂x,SwMH;w,x=∫0wMH;t,xtdt,SxMH;w,x=∫0xMH;w,ttdt,HMH;w,x=MH;w,w,QαMH;w,x=wαMH;w,x.


## 2. The Structure of Kragujevac Tree

A Kragujevac tree [22–24] is a proper tree holding a central vertex of degree at least 3, in which branches of the form Q1,Q2,Q3,… are joined. Let Q2,Q3,Q4,… be branches whose form is given in Figure 1. Kgq,r denotes the group of proper Kragujevac trees K of order q and central vertex with degree r (see details in [25–28]). If an additional vertex of degree 2 is added on one pendant line of proper Kragujevac tree, then we get a new family of Kragujevac trees named as improper Kragujevac tree, denoted by Kgq,r∗.Figure 1
Branches (a)Q2. (b) Q3. (c) Q4 (d) Q5.The branches attached to the central vertex are given byQa1,Qa2,…,Qar where ai≥2 for all i=1,…,r. In this paper, we are concerned with special case of Kragujevac tree when Qa1,Qa2,…,Qar are same (or isomorphic), i.e., a1=a2=⋯=ar=t. Hence, the order and size of Kragujevac tree K∈Kgq,r are r2t+1+1 and r2t+1, respectively. For example, see Figure 2.Figure 2
Kragujevac treeK from class Kg29,4.Graphs under inspection in this paper are undirected and simple, without various edges and finite. Considering a graphH, the sets of nodes and lines are symbolised by VH and EH, individually. Also, degree, i.e., number of lines attached to a node is denoted as dHp for node p.A huge amount of research is made on graphs, several operations on graphs, and chemical invariants since last three decades. Eliasi et al. [29] proposed operations on graphs as S,R,Q,andT named like subdivision, semitotal point graph, semitotal line graph, and total graph. Subdivision of graph H denoted by SH is derived by putting an extra vertex (mentioned as white vertex) into each edge of H. To make a contrast between already existing and newly inserted vertices, vertices of H are known as black vertices. Two white vertices will be related to each other if their correlating edges in H are adjacent and likewise two black vertices will be called related if they are adjacent in H.Subdivision and line graph of subdivision of Kragujevac tree are presented in Figures3 and 4, respectively.Figure 3
Subdivision graph of Kragujevac tree.Figure 4
Line graph of subdivision of Kragujevac tree.


## 3. Certain Topological Indices and Polynomials ofSK

In this section, we compute certain topological indices, polynomials, and several other chemical indices in terms of these polynomials as stated in the previous section for the subdivision of Kragujevac tree. The subdivision of Kragujevac tree is shown in Figure3.ForH=SK, K∈Kgq=r2t+1+1,r, forgotten and Zagreb polynomials are as follows:(32)FH,w=rwr2+4+wt2+2t+5+rtwt2+2t+5+2w8+w5,M1H,w=rwr+2+wt+3+rtwt+3+2w4+w3,M2H,w=rtw2t+2+2w4+w2+rw2t+2+rw2r,M3H,w=rtwt−1+w+2+rwt−1+rwr−2.Consider the subdivision of Kragujevac tree, denoted bySK, where K∈Kgq=r2t+1+1,r. We categorize the lines of SK according to the lines of type Edk,dp, where kt∈ESK. The line categorization is presented in Table 1.(33)FH,w=∑kp∈EHwdk2+dp2=rwr2+22+rwt+12+22+rtwt+12+22+2rtw22+22+rtw22+12=rwr2+4+wt2+2t+5+rtwt2+2t+5+2w8+w5,M1H,w=∑kp∈EHwdk+dp=∑wr+2+∑wt+1+2+∑wt+1+2+∑w2+2+∑w2+1=rwr+2+wt+3+rtwt+3+2w4+w3,M2H,w=∑kp∈EHwdkdp=rwr×2+rw2×t+1+rtw2×t+1+2rtw2×2+rtw2×1=rtw2t+2+2w4+w2+rw2t+2+rw2r,M3H,w=∑kp∈EHwdk−dp=rwr−2+rwt+1−2+rtwt+1−2+2rtw2−2+rtw2−1=rtwt−1+w+2+rwt−1+rwr−2.Table 1
Line categorization inSK, where K∈Kgq=r2t+1+1,r.
Lines of typeNumber of linesEr,2rEt+1,2rEt+1,2rtE2,22rtE2,1rtForH=SK, K∈Kgq=r2t+1+1,r, we have(34)HMH=r3+4r2+7rt2+rt3+13r+56rtRe,ReM1H=rt24t+1+7+rt22t+1+r2+1+3r2t+1Re,ReM2H=2rt2t+3+43+2rt+3t2+1+2r2r+2Re,ReM3H=2rr2+t3+4r2+8rt2+6r+48rt,M1mH=rt1t+3+56+rt+3+r1r+2R,ReM2H=r2+rt2+3rt−rRRRH,RRRH=rtt+2+t+rr−1,GO1H=rt3t+29+7r+3r2,GO2H=2rr2+t3+52rt+4r2+10rt2+6r,HGO1H=rt9t2+208+29r+12r2+39rt2,HGO2H=4rtt4+30t2+36t+170+36r+4r5+16r3+16r4,PGOH=rt12t2+4t+3+12+16+r2t2+4t+3+r2r2+4r,SGOH=rt13t+5+12+15+r3t+5+r13r+2,FH=rt3t+28+9r+rt3+r3.For line categorizations shown in Table1, using formulas of (6)–(10), (13)–(20), and (24) will give desired above expressions.ForH=SK, K∈Kgq=r2t+1+1,r, M-polynomial and certain topological indices deduced from M-polynomial are(35)MH;w,x=rwrx2+r+rtwt+1x2+2rtw2x2+rtw2x,M1H=r2+rt2+5r+15rtM2H=2r2+2rt2+14rt+2r,M2mH=r2t+1+rt12t+1+1+12Rα,RαH=r12αrα+1t+1α2α+rt1t+1α2α+22α2α+12α1α,RRαH=r2αrα+2αt+1α+rt2αt+1α+22α2α+1α2α,SDDH=rt32+6+2t+1+r12+2t+1+rt212+r22,HH=r2r+2+2t+3+rt2t+3+53,ISIH=2r2r+2+r2t+3+rt22t+3+rt4t+3+2+23,AH=r16+rt23t+13+3.For line categorizations shown in Table1, using formulas (21)–(30) will give above desired expressions.


## 4. Certain Topological Indices and Polynomials ofLSK

In this section, we calculate several topological indices, polynomials, and several other chemical indices in terms of these polynomials as mentioned in the first section for the line graph of subdivision of Kragujevac tree. The line graph of subdivision of Kragujevac tree is presented in Figure4.ForH=LSK, K∈Kgq=r2t+1+1,r, forgotten and Zagreb polynomials are as follows:(36)FH,w=r22w2r2+rur2+t2+2t+1−12w2r2+rt22w2t2+4t+2+rtw2t2+4t+22+wt2+2t+5+w5+w8,M1H,w=r2w2r2+rwr+t+1−w2r2+rt22w2t+2+rtw2t+22+wt+3+w3+w4,M2H,w=r2wr22+rwrt+r−wr22+rt22wt2+2t+1+rtwt2+2t+1+w2t+2+w2+w4,M3H,w=r212+rwr−t−1−12+rt22+rt12+wt−1+w+1.Consider the line graph of subdivision of Kragujevac tree, symbolised byLSK. We categorize the lines of LSK according to the lines of type Edk,dp, where kp∈ELSK. The line categorization is shown in Table 2.(37)FH,w=∑kp∈EHwdk2+dp2=rr−12wr2+r2+rwr2+t+12+rtt+12wt+12+t+12+rtwt+12+22+rtw22+12+rtw22+22=r22w2r2+rwr2+t2+2t+1−12w2r2+rt22u2t2+4t+2+rtw2t2+4t+22+wt2+2t+5+w5+w8,M1H,w=∑kp∈EHwdk+dp=r22wr+r−r2wr+r+rwr+t+1+rt22wt+1+t+1+rt2wt+1+t+1+rtwt+1+2+rtw2+1+rtw2+2=r2w2r2+rwr+t+1−w2r2+rt22w2t+2+rtw2t+22+wt+3+w3+w4,M2H,w=∑kp∈EHwdkdp=r22ur×r−r2wr×r+rwr×t+1+rt22ut+1×t+1+rt2wt+1×t+1+rtw2×t+1+rtw2×1+rtw2×2=r2wr22+rwrt+r−wr22+rt22wt2+2t+1+rtwt2+2t+1+w2t+2+w2+w4,M3H,w=∑kp∈EHwdk−dp=r22wr−r−r2wr−r+rwr−t−1+rt22wt+1−t−1+rt2wt+1−t−1+rtwt+1−2+rtw2−1+rtw2−2=r212+rwr−t−1−12+rt22+rt12+wt−1+w+1.Table 2
Line categorization ofLSK.
Lines of typeNumber of linesEr,rrr−1/2Et+1,rrEt+1,t+1rtt+1/2Et+1,2rtE2,1rtE2,2rtForH=LSK, K∈Kgq=r2t+1+1,r, we have(38)HMH=2r4−r3+r+38rt+2rt4+7rt3+13rt2+2r2t+2r2,ReM1H=r1+1t+1+rt22t2+2t+1+12t+2+rt31t2+2t+1+rt1t2+2t+1+rt32t+2+52+tt+1+1t+1−1,ReM2H=r314+r21r+t+1−14+rt414t+4+rt312t+2+14t+4+rt214t+4+12t+2+2t+3+rt14t+4+2t+3+53+r2tr+t+1,ReM3H=r5−r4+r3+r2+r3t+2r2t+r2t2+rt5+4rt4+8rt3+12rt2+29rt,M1mH=r14+1r+t+1+rt14t+4+1t+3+712+rt214t+4−14,RM2H=r412−r332+r232−r12+r2t+rt42+rt32+rt2,RRRH=r22r2−2r+1−r2r2−2r+1+rt32+rt22+rtrt−t+t+1,GO1H=r412+r312+r2+r+r2t+5rt32+rt42+13rt22+41rt2,GO2H=r5−r4+r3+r3t+2r2t+r2t2+rt5+4rt4+8rt3+12rt2+29rt+r2,HGO1H=r532+2r3+r+952rt2+41rt42+67rt32+9rt52+4r3t+r3t2+2r2t2+6r2t+241rt2+r62+4r2+rt62,HGO2H=2r8−2r7+r5t2+4r3t2+2r4t3+r5+4r3t2+2r4+r3+2rt8+30rt6+14rt7+22rt5+50rt4+94rt3+14rt2+328rt,PGOH=r2122r3+r1r2t+rt2+2rt+r2+r−122r3+rt12t2+8t+6+rt2122t3+6t2+6t+2+rt122t3+6t2+6t+2+16+116,SGOH=r2122r+r2+r−122r+r2+12r+rt+t+1+rt126t+2t3+2+6t2+rt15+18+13t+5+rt2126t+2t3+2+6t2,FH=r4+r+rt4+4rt3+6rt2+21rt.For line categorizations shown in Table2, use of formulas of (6)–(10), (13)–(20), and (24) will give desired above expressions. For H=LSK, K∈Kgq=r2t+1+1,r, M-polynomial and certain topological indices that can be deduced from M-polynomial are(39)MH;w,x=r22wrxr−r2wrxr+rwt+1xr+rt22wt+1xt+1+rt2wt+1xt+1+rtwt+1x2+rtw2x+rtw2x2,M1H=r3+rt3+3rt2+12rt+r,M2H=r412−r312+r2+rt412+rt332+rt272+r2t+rt172,M2mH=rt212t2+4t+2+1t+1+rt12t2+4t+2+12t+2+34+12−12r,RαH=r22rαrα−r2rαrα+rt+1αrα+rt2t+1αt+1α+rt22t+1αt+1α+rt2αt+1α+rt2α+rt2α2α,RRαH=r2rαrα2−rrαrα2+rt+1αrα+rt2t+1αt+1α2+rtt+1αt+1α2+rtt+1α2α+rt2α+rt2α2α,SDDH=r21+1t+1+rt31t+1+rt22t+1+12+rt3t+1+5+t+1,HH=r2+r2r+t+1+rt212t+2+rt12t+2+rt1t+1+rt76−12,ISIH=r44−r214+1t+r+1+rt334t+4+rt234t+4+2t+3+tr21r+t+1+rt14t+4+2t+3+53,AH=r2r3r322r−23−rr3r322r−23+rt+13r3r+t−13+rt2t+13t+1322t3+rtt+13t+1322t3+rtt+13232t+13+rt23+rt232323.For line categorizations shown in Table2, using formulas given in (21)–(30) will give desired expressions.


## 5. Numerical and Graphical Representation of Results

In this segment, numerical results were computed for different Zagreb-type indices. Moreover, Tables3–6 show numerical results of the aforementioned topological indices for various parameter values rands for the subdivision graph and for the line of subdivision graph of K where K∈Kgq=r2s+1+1,r. We have used different values of rands and compute the numerical results. Moreover, we have plotted the graphs for Kragujevac tree to study the behavior of above computed topological descriptors.Table 3
Numerical results for derived topological indices forK, K∈Kgq=r2s+1+1,r.
r,sHMHReM1HReM2HReM3HM1mHRM2HRRRH1,18275.33702.083442,23462119.86673325.0332211.4423,387611548.691810.16043.02724,419727377.0476224016.8512478.92825,53190111118.809383025.29722087.0820Table 4
Numerical results for derived topological indices forSK, where K∈Kgq=r2s+1+1,r.
r,sGO1HGO2HHGO1HHGO2HPGOHSGOHFH1,1427629710201.8162.308382,2166364139466325.16517.9321783,3403111638492545210.4514114.4974684,4732220084367656017.952124.38319965,5121041801614544718026.358935.77071870Table 5
Numerical results for derived topological indices forLSK, where K∈Kgq=r2s+1+1,r.
r,sHMHReM1HReM2HReM3HM2mHRM2HRRRH1,16675.754.33601.4133332,248219.55623.8676088.1174128.31373,3171341.562572.642833489.7143228106.633054,4486871.36170.41711296016.8087822281.42565,514630109.3048344.05533950025.40792285607.7042Table 6
Numerical results for the derived topological indices forLSK, where K∈Kgq=r2s+1+1,r.
r,sGO1HGO2HHGO1HHGO2HPGOHSGOHFH3,334602495691.56651.9816344,42166682696153524.79486.189663105,57773348161492536479.549614.974810026,621401296066732186886415.890218.903530287,749803950022380514456687524.031423.499420130Figures5–8 display different topological indices and polynomials for subdivision of Kragujevac tree.Figure 5
The graphical representation of (a)HMH, (b) ReM3H, (c) GO2H, (d) RM2H, (e) GO1H, and (f) HGO1H.Figure 6
The graphical representation of (a)ReM1H, (b) ReM2H, (c) M1mH, (d) RRRH, (e) HGO2H, and (f) PGOH.Figure 7
The graphical representation of (a)SGOH, (b) FH, (c) M1H, (d) M2H, (e) M2mH, and (f) SDDH.Figure 8
The graphical representation of (a)HH, (b) ISIH, (c) AH, (d) HMH, (e) ReM1H, and (f) ReM2H.Figures9–11 display the pictorial representation of different topological indices and polynomials for line graph of subdivision of Kragujevac tree.Figure 9
The graphical representation of (a)ReM3H, (b) M1mH, (c) RM2H, (d) RRRH, (e) GO1H, and (f) GO2H.Figure 10
The graphical representation of (a)HGO1H, (b) HGO2H, (c) PGOH, (d) SGOH, (e) FH, and (f) M1H.Figure 11
The graphical representation of (a)M2H, (b) M2mH, (c) SDDH, (d) HH, and (e) ISIH.


## 6. Conclusion

In chemical graph theory, the calculation of topological indices of any graph is significant. In this article, subdivision of graph and line graph of subdivision of Kragujevac tree is considered and we computed polynomials like Zagreb polynomials, forgotten polynomial, andM-polynomial, and also computation of chemical indices like hyper Zagreb, redefined Zagreb, modified first Zagreb, reduced second Zagreb, reduced reciprocal Randic index, family of Gourava Indices, and forgotten index is made. Further, some topological indices that can be directly derived from M-polynomial, i.e., 1st Zagreb, 2nd Zagreb, modified 2nd Zagreb and augmented Zagreb index, Randic and reciprocal Randic index, symmetric division deg index, harmonic index, and inverse sum indeg index are also computed. However, much work still needs to be done in this area.



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