# On Topological Indices of Total Graph and Its Line Graph for Kragujevac Tree Networks

Authors: Salma Kanwal; Ayesha Riasat; Muhammad Kamran Siddiqui; Shabnam Malik; Khadija Sarwar; Afzal Ammara; Asare-Tuah Anton
Journal: Complexity
 (2021)

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## Abstract

Kragujevac tree is indicated byK; K∈Kgq=s2t+1+1,s with order and size s2t+1+1 and s2t+1, respectively. In this paper, we have a look at certain topological features of the total graph and line graph of the total graph of the considered tree, i.e., Kragujevac tree, by computing different topological indices and polynomials.


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## Body

## 1. Introduction

The graphs throughout the discussion are simple. In a graphH, VH and EH denote the sets of nodes and lines accordingly. Consider dHu be the degree of a vertex u. Here, we assume that A1,A2,A3,… are the branches which are shown in Figure 1. A Kragujevac tree [1–3] with central vertex consisting of degree at least 3 is a proper tree having branches A1,A2,A3,….Figure 1
Branches (a)A2, (b) A3, (c) A4, and (d) A5.A group of proper Kragujevac trees whose central vertex consist of degrees and order q is symbolized by Kgq,s, and by adding a new vertex of degree 2 to a pendant line, an improper Kragujevac tree is attained [4]. It is symbolized by Kgq,s; see [5–7] for more details.Here, we have the branchesAl1,Al2,…,Alm where li≥2 for all i=1,…,m connected to the mid vertex. Our concern in this research work is with a specific type of Kragujevac tree when Al1,Al2,…,Alm are identical, i.e., l1=l2=⋯=lm=p. Thus, order and size of Kragujevac tree K∈Kgq,s are q=s2t+1+1 and s2t+1 accordingly, see Figure 2.Figure 2
Kragujevac treeK from class Kg36,5.Any function on a graph that does not rely on the labelling of its vertices is referred to as a graph invariant or a topological index. Many invariants have been used in QSPR/QSAR studies with varying degrees of success. Gutman and Trinajstic [8] proposed Zagreb indices three decades before. Balaban et.al. later referred to them as Zagreb group indices [9, 10].Then, the first Zagreb index and second Zagreb index are formulated as(1)M1H=∑l∈VHdl2=∑ml∈EHdm+dl,M2H=∑ml∈EHdmdl.In 2011, Fath-Tabar [11] suggested third Zagreb index for some graph H symbolized by M3H and is specifically defined as(2)M3H=∑ml∈EHdm−dl.The same author introduced Zagreb polynomials in the same year, naming them first, second, and third Zagreb polynomials, and they are formulated as(3)M1H,u=∑ml∈EHudm+dl,M2H,u=∑ml∈EHudmdl,M3H,u=∑ml∈EHudm−dl,respectively. Modified Zagreb indices [12] are another variant of Zagreb indices that are associated with degrees of vertices. The modified Zagreb indices first and second are signified by mM1H and mM2H, respectively, and are defined as(4)M1mH=∑ml∈EH1dm+dl,M2mH=∑ml∈EH1dmdl.Hyper-Zagreb index, proposed by Shirdel et.al. [13], was formulated as(5)HMH=∑ml∈EHdm+dl2.In 2013, Ranjini et.al. suggested redefined Zagreb indices [14] and were formulated as(6)redefined first Zagreb index=ReM1H=∑ml∈EHdm+dldmdl,(7)redefined second Zagreb index=ReM2H=∑ml∈EHdmdldm+dl,(8)redefined third Zagreb index=ReM3H=∑ml∈EHdmdldm+dl.Furtula et.al. introduced forgotten index which is marked byFH for graph H [15], formulated as(9)FH=∑l∈VHdl3=∑ml∈EHdm2+dl2.Parallel to the notion of forgotten topological index is that of forgotten polynomial. Forgotten polynomial of a graphH is given by(10)FH,u=∑ml∈EHudm2+dl2.Reciprocal Randić index, reduced second Zagreb index, and reduced reciprocal Randić index were proposed by Gutman et al. [16] in 2014 as follows:(11)RRH=∑ml∈EHdmdl,RM2H=∑ml∈EHdm−1dl−1,(12)RRRH=∑ml∈EHdm−1dl−1.Many new graph invariants [17–20] known as family of Gourava indices were brought into existence by V. R. Kulli in 2017 and are described as follows:(13)first Gourava index=G1OH=∑ml∈EHdm+dl+dmdl,(14)second Gourava index=G2OH=∑ml∈EHdm+dldmdl,(15)product connectivity Gourava index=PGOH=∑ml∈EH1dm+dldmdl,(16)sum connectivity Gourava index=SGOH=∑ml∈EHmdm+dl+dmdl,(17)first hyper−Gourava index=HGO1H=∑ml∈EHdm+dl+dmdl2,(18)second hyper−Gourava index=HGO2H=∑ml∈EHdm+dldmdl2.M-polynomial [21] is given as(19)MH;u,v=∑δ≤s≤p≤ΔxspHusvp,where δ and Δ are respectively minimum and maximum of vertex degrees in graph H and xsp represents the number of edges ml∈EH such that dm,dl=s,p. We can deduce various indices from M-polynomial such as(20)first Zagreb index=M1H=Du+DvMH;u,vu=v=1,(21)second Zagreb index=M2H=DuDvMH;u,vu=v=1,(22)modified second Zagreb index=M2mH=SuSvMH;u,vu=v=1,(23)Randic´index=RαH=SuαSvαMH;u,vu=v=1=∑ml∈EHdmdlα,(24)inverse Randic´index=RRH=DuαDvαMH;u,vu=v=1=∑ml∈EHdmdl.Symmetric division deg index(25)SDDH=DuSv+DvSuMH;u,vu=v=1=∑ml∈EHmindm,dlmaxdm,dl+maxdm,dlmindm,dl,(26)harmonic index=HH=2SuJMH;u,vu=v=1=∑ml∈EH2dm+dl,(27)inverse sum indeg index=ISIH=SuJDuDvMH;u,vu=v=1=∑ml∈EHdmdldm+dl,and augmented Zagreb index(28)AH=Su3Q−2Du3Dv3MH;u,vu=v=1=∑ml∈EHdmdldm+dl−23,where(29)DuMH;u,v=u∂MH;u,v∂u,DvMH;u,v=v∂MH;u,v∂v,SuMH;u,v=∫0uMH;t,vtdt,SvMH;u,v=∫0vMH;u,ttdt,JMH;u,v=MH;u,u,QαMH;u,v=uαMH;u,v.Since the past three years, people have conducted a lot of research on graphs, various operations, and chemical invariants of graphs. Observe simple charts under different chart operations such as line chart, subdivision, half-full point chart, half-full line chart, and general chart. The topological index of these operations on the graph is studied in depth.Eliasi et.al. [22] displayed operations in the chart, such as S,R,Q,T, named subdivision, semitotal dot chart, semitotal line chart, and general chart. The subdivision graph represented by SH is derived from H by replacing each of its edges or lines with a path of length 2, or in other words, by placing a new vertex (called white vertex) on each line of H [23–25]. In order to maintain the difference between the existing vertices and the newly inserted vertices, the vertices in H are called black vertices. If two white vertices are adjacent to each other in H, it is called correlation. Similarly, if two black vertices are adjacent in H, it is called correlation [26, 27].In 1981, Bertz [28] invented the first topological index established on an online graph. For these graphical operations, many authors quote [21, 29, 30] to compute various topological indices. The general diagram of the Kragujevac tree is displayed in Figure 3.Figure 3
Total graph of Kragujevac tree.Next part of this research paper comprises of two main Sections2 and 3. In Section 2, we put forward computations for several chemical invariants (through direct calculations 2.2, 3.2 and using relation between polynomials 2.1, 3.1 and chemical invariants 2.3, 3.3) for Kragujevac tree. In Section 2, graphic operation T, total graph is applied at K, whereas in Section 3, same computation is made for LTK.


## 2. Certain Topological Indices and Polynomials ofTK, where K∈Kgq=s2t+1+1,s

In this section, we compute certain topological indices, polynomials, and several other chemical indices in terms of these polynomials as mentioned in previous section for the total graph of Kragujevac tree. The edge partition is depicted in Table1.Table 1
Line distribution inTK, where K∈Kgq=s2t+1+1,s.
Lines of typeNumber of linesE2s,s+t+1sEs+t+1,2t+2sEs+t+1,t+3stEt+3,t+3stt−1/2Et+3,4stEt+3,3stE4,2stE4,3stE3,2stEs+t+1,s+t+1ss−1/2Et+3,2t+2stE2t+2,2ssE2t+2,4stTheorem 1.
LetH be the total graph of a Kragujevac tree; K∈Kgq=s2t+1+1,s. Then,(30)FH,u=∑ml∈EHudm2+dl2=su2s2+s+t+12+sus+t+12+2t+22+stus+t+12+t+32+st22ut+32+t+32−st2ut+32+t+32+stut+32+42+stut+32+32+stu42+22+stu42+32+stu32+22+s22us+t+12+s+t+12+stut+32+2t+22−s2us+t+12+s+t+12+su2t+22+2s2+stu2t+22+42,M1H,u=∑ml∈EHudm+dl=su3s+t+1+sus+t+1+2t+2+stus+t+1+t+3+st22ut+3+t+3−st2ut+3+t+3+stut+3+4+stut+3+3+stu4+2+stu4+3+stu3+2+s22us+t+1+s+t+1−s2us+t+1+s+t+1+stut+3+2t+2+su2t+2+2s+stu2t+2+4=su3s+t+1+us+3t+3−u2s+2t+22+stus+2t+4−u2t+62+ut+7+ut+6+u6+u7+u5+st2u2t+62+s2u2s+2t+22+stu3t+5+su2t+2+2s+stu2t+6,M2H,u=∑ml∈EHudmdl=su2s×s+t+1+sus+t+1×2t+2+stus+t+1×t+3+st22ut+3×t+3−st2ut+3×t+3+stut+3×4+stut+3×3+stu3×2+su2t+2×2s+s22us+t+1×s+t+1−s2us+t+1×s+t+1+stut+3×2t+2+stu2t+2×4+stu4×2+stu4×3+stut+3×4=su2s2+2st+2s+u2st+4t+2s+2t2+2−us2+2st+2s+2t+t2+12+stut2+st+4t+3s+3+ut2+6t+92+u12+u4t+12+u3t+9+u8+u6+st2ut2+6t+92+s2us2+2st+2s+2t+t2+12+stu2t2+8t+6+su4st+4s+stu8t+8,M3H,u=∑ml∈EHudm−dl=su2s−s+t+1+sus+t+1−2t+2+stus+t+1−t+3+ts22ut+3−t+3−st2ut+3−t+3+stut+3−4+stut+3−3+stu4−2+stu4−3+stu3−2+s22us+t+1−s+t+1−s2us+t+1−s+t+1+stut+3−2t+2+su2t+2−2s+stu2t+2−4=s2us−t−1−12+stus−2−12+ut−1+ut+u2+2u+ts212+s212+stu1−t+su2t+2−2s+stu2t−2.Proof.
Consider the total graph of Kragujevac tree, denoted byTK, where K∈Kgq=s2t+1+1,s. We categorize the lines of TK based on the lines of type Edm,dl, where ml∈ETK. The line classification is shown in Table 1,(31)FH,u=∑ml∈EHudm2+dl2=su2s2+s+t+12+sus+t+12+2t+22+stus+t+12+t+32+st22ut+32+t+32−st2ut+32+t+32+stut+32+42+stut+32+32+stu42+22+stu42+32+stu32+22+s22us+t+12+s+t+12+stut+32+2t+22−s2us+t+12+s+t+12+su2t+22+2s2+stu2t+22+42,M1H,u=∑ml∈EHudm+dl=su3s+t+1+sus+t+1+2t+2+stus+t+1+t+3+st22ut+3+t+3−st2ut+3+t+3+stut+3+4+stut+3+3+stu4+2+stu4+3+stu3+2+s22us+t+1+s+t+1−s2us+t+1+s+t+1+stut+3+2t+2+su2t+2+2s+stu2t+2+4=su3s+t+1+us+3t+3−u2s+2t+22+stus+2t+4−u2t+62+ut+7+ut+6+u6+u7+u5+st2u2t+62+s2u2s+2t+22+stu3t+5+su2t+2+2s+stu2t+6,M2H,u=∑ml∈EHudmdl=su2s×s+t+1+sus+t+1×2t+2+stus+t+1×t+3+st22ut+3×t+3−st2ut+3×t+3+stut+3×4+stut+3×3+stu3×2+su2t+2×2s+s22us+t+1×s+t+1−s2us+t+1×s+t+1+stut+3×2t+2+stu2t+2×4+stu4×2+stu4×3+stut+3×4=su2s2+2st+2s+u2st+4t+2s+2t2+2−us2+2st+2s+2t+t2+12+stut2+st+4t+3s+3+ut2+6t+92+u12+u4t+12+u3t+9+u8+u6+st2ut2+6t+92+s2us2+2st+2s+2t+t2+12+stu2t2+8t+6+su4st+4s+stu8t+8,M3H,u=∑ml∈EHudm−dl=su2s−s+t+1+sus+t+1−2t+2+stus+t+1−t+3+ts22ut+3−t+3−st2ut+3−t+3+stut+3−4+stut+3−3+stu4−2+stu4−3+stu3−2+s22us+t+1−s+t+1−s2us+t+1−s+t+1+stut+3−2t+2+su2t+2−2s+stu2t+2−4=s2us−t−1−12+stus−2−12+ut−1+ut+u2+2u+ts212+s212+stu1−t+su2t+2−2s+stu2t−2.Theorem 2.
LetH be the total graph of Kragujevac tree; K∈Kgq=s2t+1+1,s, then we have(32)HMH=12s3+56st2+8s+209st+20s2t+10s2+5s3t+16st3+6s2t2+2st4+2s4+13st3+69st+58st2+8st2+4s+4s3+8s2,ReM1H=s32s+2t+2+32st+4t+2s+2t2+2−1s2+2st+2s+2t+t2+1+st32st+2s+2t2+4t+2+4t2+st+4t+3s+3−3t2+6t+9+74t+12+2t+3+14+712+56−1s2+2st+2s+2t+t2+1+s2t1t2+st+4t+3s+3+1s2+2st+2s+2t+t2+1+st22t2+st+4t+3s+3+3t2+6t+9−1t2+6t+9+14t+12+13t+9+st31t2+6t+9+st2t+68t+8+t12t+2s+2+s31s2+2st+2s+2t+t2+1+st3t+52t2+8t+6+s2t+2+2s4st+4s,ReM2H=2s33s+t+1+2s2t3s+t+1+2s23s+t+1+2s2ts+3t+3+4sts+3t+3+2s2s+3t+3+2st2s+3t+3+2ss+3t+3+st3s+2t+4+s2t2s+2t+4+4st2s+2t+4+3s2ts+2t+4+3sts+2t+4+st44t+12+3st32t+6+9st24t+12−st34t+12−6st24t+12−9st4t+12+4st2t+7+12stt+7+3st2t+6+9stt+6+4st3+12st7+6st5+s44s+4t+4+s3t2s+2t+2+s32s+2t+2+s2t2s+2t+2+s2t24s+4t+4+s24s+4t+4−s34s+4t+4−s2t2s+2t+2−s22s+2t+2−st2s+2t+2−st24s+4t+4−s4s+4t+4+st2t2+8t+6t+3+2t+2+s4st+4s2t+2+2s+st8t+82t+6,ReM3H=s5+s46+s38−s28+s5+36s2t+104st2+24s2t2+4s2t3+13s3t+42st3+4s3t2+2s3+6st4+50st3+122st2+78st+16s2t+8s2t2+8s2+8s3,M1mH=s13s+t+1+1s+3t+3−14s+4t+4+st1s+2t+4−14t+12+1t+7+1t+6+16+17+15+st214t+12+s214s+4t+4+st13t+5+s12t+2+2s+st12t+6,(33)RM2H=s42+3s32+2s2−s+s3t+9s2t+22st+39st22+3s2t22+9st32+st42,RRRH=s2s2+2st−s−t+2st+2t2+s+t−s2+2st+t22+stst+2s+t2+2t−t2+4t+42+3t+6+2t+4+3+6+2+st2t2+4t+42+s2s2+2st+t22+stt+22t+1+s2t+12s−1+st2t+13,PGOH=s16s3+8st2+8s2t+4st+2s+2st2+s12s2t+8st2+10st+24t2+6t3+18t+2s+6+st13st2+2t3+12t2+14st+16t+3s2+15s+18+st2212t3+18t2+58t+54+s12t+2+2s4st+4s−st212t3+18t2+58t+54+st14t2+40t+84+st148+st184+s2212s3+6s2t+6st2+12st+8s2+6s+4t2+2t3+2+6t−s212s3+6s2p+6st2+12st+8s2+6s+4t2+2t3+2+6t+st13t2+27t+54+st13t+52t2+11t+11+s12t+2+2s4st+4s+st12t+68t+8+st13t+52t2+8t+6+st12t+68t+8+st130,GO1H=s5s+2s2+2st+t+1+s3s+7t+2st+2t2+5+sts+2t+4+st+t2+t+3s+3t+1+stt−12t2+8t+15+st5t+19+st4t+15+st14+st21+st11+ss−124s+4t+3+s2+t2+2st+st2t2+11t+11+s4st+2t+2+6s+st10t+14,GO2H=s5+s46+s38+s5+52s2t+226st2+32s2t2+4s2t3+29s3t+92st3+4s3t2+2s3+78st+8s2,HGO1H=37s3+12s2t+5s2+24s3t+st2+4s2t2+10s4+4s5+6s4t+2s3t2+9s3+62s2t+40s2t2+12s3t+30s2+69st2+28st3+24st4+8s2t2+35st+25s+st5+2s2t4+6st4+14st3+s3t3+8s3t2+31s2t2+16s3t+63st2+42s2t+49st+st62+8st5+47st4+120st3+225st22−st52−8st4−47st3−120st2−225st2+25st3+190st2+361st+121st+s62+3s4t2+3s5t+4s5+12s4t+11s4+27s2t4+2s2t4+2s3t3+12s3t2+11s2t2+22s3t+2s2t3+12s2t+6s3+9t22−s52−3s3t2−3s4t−4s4−12s3t+11s2t3−11t3−5st42−2s3t3−12s2t2−11st2−22s2t−2st3−12st−6s2−9t2,HGO2H=s6s3+8s2+8s2t+4st+2s+2st22+s2s2t+2s2+16st+8st2+18t2+6t3+18t+8s+62+st3st2+2t3+12t2+14st+s2t+15s+12+12t2+st222t3+18t2+54t+542−st22t3+18t2+54t+542+st4t2+40t+842+st3t2+27t+542+st482+st842+st302+s222s3+6s2t+6st2+12st+6s2+6s+6t2+2t3+2+6t2−s22t3+6s2t+6st2+12st+6s2+6s+6t2+2t3+2+6t2,(34)SGOH=s13s+t+1+2s2+2st+2s+s1s+3t+3+2s+4t+2t2+2+st221t2+6t+9+2t+6−st2t2+6t+9+2t+6+st15t+12+7+st14t+15+st114+st119+st111+st110t+14+s221s2+2st+2s+2t+t2+1+2s+2t+2−s21s2+2st+2s+2t+t2+1+2s+2t+2+st12t2+11t+11+s14st+2t+2+6s+st1t2+st+4t+3s+3+s+4,FH=4s3+s3+st2+s+2st+2s2t+2s2+s3+st2+s+2st+2s2t+2s2+4st2+4s+8st+s3t+st3+st+2st2+2s2t+2s2t2+st3+9st+6st2+st42+9st22+3st3+st42+9st22+3st3−st32−3st2−9st2−3st2−st32−9st2+st3+9st+6st2+16st+st3+9st+6st2+9st+20st+28st+13st+s42+st222+s22+s2t+s3+s3t−s32−st22−s2−st−s2−s32−st22−2st−st2−s2−s2t+st5t2+14t+13+s4t2+4+8t+4s2+st4t2+20+8t.Proof.
For line classification shown in Table1, use of formulas (5)–(9), (11)–(18), and (22) will give desired above expressions.Theorem 3.
LetH be the total graph of a Kragujevac tree; K∈Kgq=s2t+1+1,s, then M-polynomial and certain topological indices deducted from M-polynomial are(35)MH;u,v=su2svs+t+1+sus+t+1v2t+2+stus+t+1vt+3+st22ut+3vt+3−st2ut+3vt+3+stut+3v4+stut+3v3+stu4v2+stu4v3+stu3v2+s22us+t+1vs+t+1−s2us+t+1vs+t+1+stut+3v2t+2+su2t+2v2s+stu2t+2v4,M1H=6s2+952st+5s+2s2t+11st2+st3+s3,M2H=5s32+7s2t+7s22+14st2+97st2+32s+3s2t22+7st32+st42+s42+s3+16st2+14st+4s2+2st3+4s2t,M2mH=12s+2t+2+s12t2+2st+4t+2s+2−12s+t+12+st1t2+st+4t+3s+3−12t2+2t+18+14t+12+st12t+24+st8+13t+9+18+112+16+st212t2+12t+18+s212s+t+12+st12t+2t+3+s12t+22s,RαH=s2sαs+t+1α+s2t+2αs+t+1α+stt+3αs+t+1α+st22t+3αt+3α−st2t+3αt+3α+st4αt+3α+st3αt+3α+st3α4α+st2α3α+st4α2α+s22s+t+1αs+t+1α−s2s+t+1αs+t+1α+st12t+2αt+3α+s12sα2t+2α+st14α2t+2α,(36)RRαH=s2sαs+t+1α+ss+t+1α2t+2α+sts+t+1αt+3α−s2s+t+1αs+t+1α+st22t+3αt+3α−st2t+3αt+3α+stt+3α4α+stt+3α3α+st4α2α+st4α3α+st3α2α+s22s+t+1αs+t+1α+st2t+2αt+3α+s2sα2t+2α+st4α2t+2α,SDDH=s22s+t+1+12t+2+1+st12t+2+2t+3+2s+t+1+32t+3+294+s2t1t+3+12t+2+st21t+3+1s+t+1+1t+3+1312+st312t+3+12+t2+s1s+t+1+12+2st,HH=s23s+t+1+2s+3t+4−12s+2t+2+st2s+2t+4−12t+6+2t+7+2t+6+13+27+25+s212s+2t+2+st212t+6+st23t+5+s22t+2+2s+st22t+6,ISIH=s2ss+t+113s+t+1+ss+t+12t+21s+3t+3+sts+t+1t+31s+2t+4+st22t+3t+312t+6+st4317+st3215−st2t+3t+312t+6+stt+341t+7+stt+331t+6+st4216+s22s+t+1s+t+112s+2t+2−s2s+t+1s+t+112s+2t+2+stt+32t+23t+5+s2t+22s2t+2s+2+st2t+242t+6,AH=s2s3s+t+1313s+t−13+ss+t+132t+231s+3t+13+sts+t+13t+331s+2t+23+st22t+33t+3312t+43−st2t+33t+3312t+43+stt+33431t+53+stt+33331t+43+st4323143+st4333153+st3323133+s22s+t+13s+t+1312s+2t3−s2s+t+13s+t+1312s+2t3+st2t+232t+3313t+33+s2s32t+2312t+2s3+st432t+2312t+43.Proof.
For line distribution shown in Table1, use of formulas (19)–(28) will give above desired expressions.


## 3. Certain Topological Indices and Polynomials ofLTK, where K∈Kgq=s2t+1+1,s

In this section, we calculate specific topological indices and polynomials and many other chemical indices in terms of these polynomials, as presented in the first section of the Kragujevac tree total line graph. Figures4–7 represents certain topological indices and polynomials.Figure 4
The pictorial representation of (a)HMH, (b) ReM1H, (c) ReM2H, (d) ReM3H, (e) M1mH, and (f) RM2H in space.Figure 5
The pictorial characterization of (a)RRRH, (b) GO1H, (c) GO2H, (d) HGO1H, (e) HGO2H, and (f) PGOH in space.Figure 6
The pictorial representation of (a)SGOH, (b) FH, (c) M1H, (d) M2H, (e) M2m, and (f) SDDH in space.Figure 7
The pictorial characterization of (a)HH and (b) ISIH in space.Theorem 4.
LetH be the line graph of total of Kragujevac tree; K∈Kgq=s2t+1+1,s. Then,(37)FH,u=∑ml∈EHudm2+dl2=stut+52+t+42+stu2t+42+t+52+stut+52+2t+s+22+stut+52+3t+32+sts−1ut+52+2t+42+sts−1ut+42+2t+42+stu42+52+stut+42+3t+32+stut+42+2t+s+22+stut+42+52+stut+42+32+stut+52+52+stu52+32+stu2t+s+32+3t+32+stt−1u3t+32+2t+42+stt−1u2t+s+22+2t+42+stt−1t−22u2t+42+2t+42+stu42+32+stu42+t+52+stu2t+42+42+stu2t+42+52+stu2t+s+22+s+3t+12+stt−12u2t+s+22+2t+s+22+stu3s+t−12+2t+s+22+su3s+t−12+s+3t+12+sts−1u2t+s+22+2s+2t2+ss−1s−22u2s+2t2+2s+2t2+ss−1u3s+t−12+2s+2t2+ss−1us+3t+12+2s+2t2+ss−1u2s+2t2+2s+2t2+ss−12u3s+t−12+3s+t−12+ss−1u2s+2t2+3s+t−12+s−1u2s+2t2+2s+2t2+stu2t+42+s+3t+12+stu2t+42+2s+2t2+stt−12u3t+32+3t+32+stu3t+32+s+3t+12+stu3t+32+2s+2t2+sus+3t+12+2s+2t2,M1H,u=∑ml∈EHudm+dl=stut+5+t+4+stu2t+4+t+5+stut+5+2t+s+2+stut+5+3t+3+sts−1ut+5+2t+4+sts−1ut+4+2t+4+stut+4+3t+3+stut+4+2t+s+2+stut+4+5+stut+4+3+stut+5+5+stu5+3+stu2t+s+3+3t+3+stt−1u3t+3+2t+4+stt−1u2t+s+2+2t+4+sus+3t+1+2s+2t+stt−1t−22u2t+4+2t+4+stu4+3+stu4+5+stu4+t+5+stu2t+4+4+stu2t+4+5+stu2t+s+2+s+3t+1+stt−12u2t+s+2+2t+s+2+stu3s+t−1+2t+s+2+su3s+t−1+s+3t+1+sts−1u2t+s+2+2s+2t+ss−1s−22u2s+2t+2s+2t+ss−1u3s+t−1+2s+2t+ss−1us+3t+1+2s+2t+ss−1u2s+2t+2s+2t+ss−12u3s+t−1+3s+t−1+ss−1u2s+2t+3s+t−1+s−1u2s+2t+2s+2t+stu2t+4+s+3t+1+stu2t+4+2s+2t+stt−12u3t+3+3t+3+stu3t+3+s+3t+1+stu3t+3+2s+2t,(38)M2H,u=∑ml∈EHudmdl=stut+5×t+4+stu2t+4×t+5+stut+5×2t+s+2+stut+5×3t+3+sts−1ut+5×2t+4+sts−1ut+4×2t+4+stut+4×3t+3+stut+4×2t+s+2+stut+4×5+stut+4×3+stut+5×5+stu5×3+stu2t+s+3×3t+3+stt−1u3t+3×2t+4+stt−1u2t+s+2×2t+4+stt−1t−22u2t+4×2t+4+stu4×3+stu4×5+stu4×t+5+stu2t+4×4+stu2t+4×5+stu2t+s+2×s+3t+1+stt−12u2t+s+2×2t+s+2+stu3s+t−1×2t+s+2+su3s+t−1×s+3t+1+sts−1u2t+s+2×2s+2t+ss−1s−22u2s+2t×2s+2t+ss−1u3s+t−1×2s+2t+ss−1us+3t+1×2s+2t+ss−1u2s+2t×2s+2t+ss−12u3s+t−1×3s+t−1+ss−1u2s+2t×3s+t−1+s−1u2s+2t×2s+2t+stu2t+4×s+3t+1+stu2t+4×2s+2t+stt−12u3t+3×3t+3+stu3t+3×s+3t+1+stu3t+3×2s+2t+sus+3t+1×2s+2t,M3H,u=∑ml∈EHudm−dl=stut+5−t+4+stu2t+4−t+5+stut+5−2t+s+2+stut+5−3t+3+sts−1ut+5−2t+4+sts−1ut+4−2t+4+stut+4−3t+3+stut+4−2t+s+2+stut+4−5+stut+4−3+stut+5−5+stu5−3+stu2t+s+3−3t+3+stt−1u3t+3−2t+4+stt−1u2t+s+2−2t+4+stt−1t−22u2t+4−2t+4+stu4−3+stu4−5+stu4−t+5+stu2t+4−4+stu2t+4−5+stu2t+s+2−s+3t+1+stt−12u2t+s+2−2t+s+2+stu3s+t−1−2t+s+2+su3s+t−1−s+3t+1+sts−1u2t+s+2−2s+2t+ss−1s−22u2s+2t−2s+2t+ss−1u3s+t−1−2s+2t+ss−1us+3t+1−2s+2t+ss−1u2s+2t−2s+2t+ss−12u3s+t−1−3s+t−1+ss−1u2s+2t−3s+t−1+s−1u2s+2t−2s+2t+stu2s+2t−s+3t+1+stu2t+4−2s+2t+stt−12u3t+3−3t+3+stu3t+3−s+3t+1+stu3t+3−2s+2t+sus+3t+1−2s+2t.Proof.
Consider the line graph of total graph of Kragujevac tree, denoted byLTK. We categorize the lines of LTK based on the lines of type Edm,dl, where ml∈ELTK. The line distribution is shown in Table 2.(39)FH,u=∑ml∈EHudm2+dl2=stut+52+t+42+stu2t+42+t+52+stut+52+2t+s+22+stut+52+3t+32+sts−1ut+52+2t+42+sts−1ut+42+2t+42+stu42+52+stu2t+42+3t+32+stut+42+2t+s+22+stut+42+52+stut+42+32+stut+52+52+stu52+32+stu2t+s+32+3t+32+stt−1u3t+32+2t+42+stt−1u2t+s+22+2t+42+stt−1t−22u2t+42+2t+42+stu42+32+stu42+t+52+stu2t+42+42+stu2t+42+52+stu2t+s+22+s+3t+12+stt−12u2t+s+22+2t+s+22+stu3s+t−12+2t+s+22+su3s+t−12+s+3t+12+sts−1u2t+s+22+2s+2t2+ss−1s−22u2s+2t2+2s+2t2+ss−1u3s+t−12+2s+2t2+ss−1us+3t+12+2s+2t2+ss−1u2s+2t2+2s+2t2+ss−12u3s+t−12+3s+t−12+ss−1u2s+2t2+3s+t−12+s−1u2s+2t2+2s+2t2+stu2t+42+s+3t+12+stu2t+42+2s+2t2+stt−12u3t+32+3t+32+stu3t+32+s+3t+12+stu3t+32+2s+2t2+sus+3t+12+2s+2t2,M1H,u=∑ml∈EHudm+dl=stut+5+t+4+stu2t+4+t+5+stut+5+2t+s+2+stut+5+3t+3+sts−1ut+5+2t+4+sts−1ut+4+2t+4+stut+4+3t+3+stut+4+2t+s+2+stut+4+5+stut+4+3+stut+5+5+stu5+3+stu2t+s+3+3t+3+stt−1u3t+3+2t+4+stt−1u2t+s+2+2t+4+sus+3t+1+2s+2t+stt−1t−22u2t+4+2t+4+stu4+3+stu4+5+stu4+t+5+stu2t+4+4+stu2t+4+5+stu2t+s+2+s+3t+1+stt−12u2t+s+2+2t+s+2+stu3s+t−1+2t+s+2+su3s+t−1+s+3t+1+sts−1u2t+s+2+2s+2t+ss−1s−22u2s+2t+2s+2t+ss−1u3s+t−1+2s+2t+ss−1us+3t+1+2s+2t+ss−1u2s+2t+2s+2t+ss−12u3s+t−1+3s+t−1+ss−1u2s+2t+3s+t−1+s−1u2s+2t+2s+2t+stu2t+4+s+3t+1+stu2t+4+2s+2t+stt−12u3t+3+3t+3+stu3t+3+s+3t+1+stu3t+3+2s+2t,(40)M2H,u=∑ml∈EHudmdl=stut+5×t+4+stu2t+4×t+5+stut+5×2t+s+2+stut+5×3t+3+sts−1ut+5×2t+4+sts−1ut+4×2t+4+stut+4×3t+3+stut+4×2t+s+2+stut+4×5+stut+4×3+stut+5×5+stu5×3+stu2t+s+3×3t+3+stt−1u3t+3×2t+4+stt−1u2t+s+2×2t+4+stt−1t−22u2t+4×2t+4+stu4×3+stu4×5+stu4×t+5+stu2t+4×4+stu2t+4×5+stu2t+s+2×s+3t+1+stt−12u2t+s+2×2t+s+2+stu3s+t−1×2t+s+2+su3s+t−1×s+3t+1+sts−1u2t+s+2×2s+2t+ss−1s−22u2s+2t×2s+2t+ss−1u3s+t−1×2s+2t+ss−1us+3t+1×2s+2t+ss−1u2s+2t×2s+2t+ss−12u3s+t−1×3s+t−1+ss−1u2s+2t×3s+t−1+s−1u2s+2t×2s+2t+stu2t+4×s+3t+1+stu2t+4×2s+2t+stt−12u3t+3×3t+3+stu3t+3×s+3t+1+stu3t+3×2s+2t+sus+3t+1×2s+2t,M3H,u=∑ml∈EHudm−dl=stut+5−t+4+stu2t+4−t+5+stut+5−2t+s+2+stut+5−3t+3+sts−1ut+5−2t+4+sts−1ut+4−2t+4+stut+4−3t+3+stut+4−2t+s+2+stut+4−5+stut+4−3+stut+5−5+stu5−3+stu2t+s+3−3t+3+stt−1u3t+3−2t+4+stt−1u2t+s+2−2t+4+stt−1t−22u2t+4−2t+4+stu4−3+stu4−5+stu4−t+5+stu2t+4−4+stu2t+4−5+stu2t+s+2−s+3t+1+stt−12u2t+s+2−2t+s+2+stu3s+t−1−2t+s+2+su3s+t−1−s+3t+1+sts−1u2t+s+2−2s+2t+ss−1s−22u2s+2t−2s+2t+ss−1u3s+t−1−2s+2t+ss−1us+3t+1−2s+2t+ss−1u2s+2t−2s+2t+ss−12u3s+t−1−3s+t−1+ss−1u2s+2t−3s+t−1+s−1u2s+2t−2s+2t+stu2t+4−s+3t+1+stu2t+4−2s+2t+stt−12u3t+3−3t+3+stu3t+3−s+3t+1+stu3t+3−2s+2t+sus+3t+1−2s+2t.Table 2
Line classification ofLTK.
Lines of typeNumber of linesEt+5,t+4stE2t+4,t+5stEt+5,2t+s+2stEt+5,3t+3stEt+5,2t+4sts−1Et+4,2t+4sts−1Et+4,3t+3stEt+4,2t+s+2stEt+4,5stEt+4,3stEt+5,5stE5,3stE2t+s+2,3t+3stE3t+3,2t+4sts−1E2t+s+2,2t+4sts−1E2t+4,2t+4stt−1t−2/2E4,3stE4,5stE4,t+5stE2t+4,4stE2t+4,5stE2t+s+2,2t+s+2stt−1/2E2t+s+2,s+3t+1stE3s+t−1,2t+s+2stE3s+t−1,s+3t+1sE2t+s+2,2s+2tsts−1E2s+2t,2s+2tss−1s−2/2E3s+t−1,2s+2tss−1Es+3t+1,2s+2tss−1E2s+2t,2s+2tss−1E3s+t−1,3s+t−1ss−1/2E2s+2t,3s+t−1ss−1E2s+2t,2s+2ts−1E2t+4,s+3t+1stE2t+4,2s+2tstE3t+3,3t+3stt−1/2E3t+3,s+3t+1stE3t+3,2s+2tstEs+3t+1,2s+2tsTheorem 5.
LetH be the line graph of total of Kragujevac tree; K∈Kgq=s2t+1+1,s, then we have(41)HMH=74st4+248st3+522st2+830st+173s2t+293s2t2+50s2t3+36s3t2+8st5−8s+133s3t−115s3+26s2,ReM1H=st22t2+9t+20+32t2+14t+20+32t2+st+12t+5t+14+42t2+st+12t+5s+14−32t2+14t+20+32t2+12t+16+43t2+15t+12+15t+25+56t2+12t+3rt+3s+6+76t2+18t+12−56t2+18t+12+32t2+6t+st+2s+4−22t2+6t+st+2t+4−14t2+12t+2st+4s+8+14t+20+14t+8+15t+10+56t2+6t+s2+16st+3s+24t2+4st+8t+s2+8s+4−24t2+4st+8t+s2+8s+4+37st+3s2+5s+2t2−2−23st+2t2+s2+2s+2t+52rt+6t2+14t+4s+4+1t2+st+2t+2s+13t2+6t+3−13t2+6t+3+13t2+4t+s+st+4+56t2+6st+6t+6s+st9t2+9t+20+92t2+14t+2t+72t2+st+12t+5s+4+82t2+st+12t+5s+4−92t2+14t+20−4t2+6t+8+73t2+15t+12+2t+5+815+56t2+12t+3st+3t+6−76t2+18t+12−32t2+6t+st+2s+4+712+920+94t+20+1t+2+910t+20+36st+6t2+8t+s2+3s−24t2+4st+8t+s2+8s+4+17st+3s2+5s+2t2−2+47st+3s2+5s+2t2−2−13st+2t2+s2+2t+2s+1s2+t2+2st−36s2+2t2+8st−2s−2t−52s2+8st+2s+2t+6t2−1s2+t2+2st−19s2+t2+6st+1−2t−6s−36s2+2t2+8st−2t−2s+1s2+t2+2st+56t2+2st+14t+4s+4+1s2+st+2t+2s−13t2+6t+3+49t2+12t+3t+3st+3+12t2+2st+2t+2s+56t2+2s2+8st+2t+2s+s2t12t2+st+12t+5s+14+92t2+14t+20+4t2+6t+8+16t2+12t+3st+3t+6(42)−14t2+12t+2st+4s+8+26st+6t2+8t+s2+3s−14t2+4st+8t+s2+8s+4+47st+3s25s+2t2−2+13st+2t2+s2+2s+2t−36st+4t2+2s2+4s+4t−32s2+2t2+4st+36s2+2t2+8st+2s−2t+52s2+8st+2s+2t+6t2+1t2+s2+2st+19s2+t2+1+6st−2t−6s+38st+6s2+2t2−2s−2t+12t2+2st+4t+4s+19t2+12t+3s+3st+3+12st+6t2+14t+4s+4+13t2+3st+3t+3s+s2t232t2+14t+20+32t2+12t+16+14t2+12t+2st+4s+8+14t2+4st+8t+s2+8s+4+23st+2t2+s2+2s+2t+st356t2+18t+12+44t2+12t+2st+4s+8+24t2+4st+8t+s2+8s+4+13t2+6t+3+s247st+3s2+5s+2t2−2+1s2+t2+2st−56s2+2t2+8st+2s−2t−16s2+2t2+8st−2s−2t+12s2+8st+2s+2t+6t2(43)−1t2+r2+2st−19s2+t2+1+6st−2t−6s−39s2+t2+1+6st−2t−6s−18st+6s2+2t2−2t−2s−58st+6s2+2t2−2t−2s+1t2+s2+2st+36t2+2s2+8st+2t+2s+s3t36st+4t2+2s2+4s+4t+12s2+2t2+4st+s412s2+2t2+4st+s3−32s2+2t2+4st+56s2+2t2+8s−2s−2t+32s2+8st+2s+2t+6t2+1t2+s2+2st+39s2+t2+6st−2t−6s+58st+6s2+2t2−2t−2s+s16s2+2t2+8st−2s−2t−12s2+8st+2s+2t+6t2−19s2+t2+1+6st−2t−6s+18st+6s2+2t2−2t−2s−1s2+t2+2st+16t2+2s28st+2s+2t+s−1t2+s2+2st,ReM2H=st312t+9+23t+9+23t+s+7+34t+8−23t+8+34t+7+23t+s+6+65t+s+5+185t+7−65t+7+124t+s+6+2t+2+12t+4−4t+2+65t+2s+3+22t+s+2−12t+s+2+24s+3t+1−43s+4t+2+65t+s+5+22t+s+2+32t+2−34t+4+96t+s+4+65t+2s+3+st292t+9+143t+9+23t+s+7+122t+s+7+92t+4−123t+8+154t+7+103t+s+6+5t+9+3t+7+5t+10+125t+s+5+125t+7−185t+7+84t+s+6−22t2+6t+st+2s+4+2t+2−4t+2+4t+9+4t+4+102t+9+85t+2s+3+12t+s+2−22t+s+2+74s+4t−43s+t+2−1s+t−25s+3t−1−63s+5t+1−1s+t−112s+4t−4−25s+3t−1+1s+t+145t+s+5+42t+s+2+34t+4−32t+2+126t+s+4+65t+2s+3+63s+5t+1+st202t+9+203t+9+142t+s+7+154t+8−163t+8+124t+7+83t+s+6+20t+9+12t+7+25t+10+158+65t+s+5−125t+7−32t2+6t+st+2s+4+2t+2+127+209+20t+9+8t+4+102t+9−12t+s+2−24s+3t+1+45t+s+5−12s+2t+25s+3p−1−23s+5p+1−36s+2t−2+25s+3t−1−2s+t−32t+4+36t+s+4+23s+5t+1+s2t213t+s+7+4t+3+13t+s+6+35t+s+5+44t+s+6+63s+5t+1+1s+t+112s+4t+4+25s+3t−1+25t+s+5+22t+s+2+65t+2s+3+36t+s+4+s2t53t+r+7+163t+9+43t+s+6+35t+s+5−14t2+12t+2st+4s+8+36t+s+4+35t+2s+3−22t+s+2+54s+3t+1+52s+2t−43s+4t+2+2s+t−25s+3t−1−85s+3t−1+23s+5t+1−83s+5t+1−2s+t−16s+2t−2+36s+2t−2−85s+3t−1−25s+3t−1+2s+t+45t+s+5+42t+s+2+65t+2s+3+83s+5t+1+s2t323t+9+24t+s+6+12t+s+2+43s+4t+2(44)+st465t+7+44t+s+6+2t+2+14t+4−1t+2+12t+s+2+st512t+4+s3t15t+2s+3−18t+4s+8+34s+3t+1+43s+4t+2−23s+4t+2−3s+t+85s+3t−1+83s+5t+1+2s+t+36s+2t−2+85s+3t−1+s3t218t+4r+8+63s+4t+2+12s+2t+s334s+4t+1s+t−25s+3t−1−65s+3t−1+23s+5t+1−23s+5t+1−1s+t−36s+2t−2−912s+4tt−4−25s+3t−1−65s+3t−1+1s+t+23s+5t+1+s212s+2t+25s+3t−1−23s+5t+1+112s+4t−4−36s+2t−2+25s+3t−1+23s+5t+1+s−14s+4t−112s+4t−4+s4t23s+4t+2+1s+t+s512s+2t+s4−32s+2t+65s+3t−1+23s+5t+1+1s+t+912s+4t−4+65s+3t−1+t2−1s+t,ReM3H=st143t3+406t2+1000t+1745+141st2+48s2+304st+128s+50s2t+7s3+s2t28t3+136t2+338t+308+6s3+26s2t+36st2+16s2+40st+8s+st265t3+213t2+203t+39+8s2t+52s+14s2+24st2+72st+2s3+st38t3+48t2+96t+64+s−59s3+15t3+9s2t+67s2+s29st2+11t2−17s−11t+66st+s38t3+8s2+24st2+24s2t+s25t3+85s3+105st2+153s2t−51s2−3t2−4st+9t+15s−2,M1mH=st12t+9+13t+9+13t+s+7+14t+8+13t+9−13t+8+14t+7+13t+s+6+1t+9+1t+7+1t+10+18+15t+s+5−15t+7−14t+s+6+124t+8+17+19+1t+9+12t+8+12t+9+15s+2t+3−14t+2s+4+14s+3t+1−13s+4t+2+15t+s+5+14t+2s+4−16t+6+16t+s+4+15t+2s+3+s2t13t+9+13t+8+15t+7+13s+4t+2+st214t+s+6−14t+8+124t+2s+4+16t+6+st314s+4t+14s+4t−15s+3t−1(45)−13s+5t+1−14s+4t−126s+2t+2−15s+3t−1+14s+4t+13s+5t+1+s2−324s+4t+15s+3t+1+13s+5t+1+14s+4t+126s+2t−2+15t+3t−1+s314s+4t−14s+4t,RM2H=st−512t2+53t+102+6st+11s+s2+s2t8t2+20+2s2+8st+s+st2252t2+17t−112+6st+4s+s22+st324t2+12t+9+s−19s22−11t22−5st+30s+21t−12+s32s2+2t2+4st−2s−2t+12+s233s22+17t22+17st−19s−13t+112,RRRH=stt2+12+7t+2t2+11t+12+3t2+14t+8+2t2+9t+4s+st+4−2t2+11t+12−2t2+7t+3s+st+3+2t2+9t+9+3t2+11t+6+4t+12+2t+6+4t+16+8+6t2+7t+2s+3st+2−6t2+13t+6−4t2+8t+3s+4st+1+14t2+12t+9/22+6+12+3t+12+6t+9+8t+12+6t2+5st+s2+3t+s,GO1H=st111t22+184t+6032+55s+23st+2s2+s2t6t2+40t+55+6st+3s2+4s+st229t22+51t+812+5s+2st+s5t22−19s22−4s−2t−st+32+s32s+2t+2s2+2t2+4st+s217t22+33s22+9s+7t+23st−12−4t+4s+14t2+4s2+8st,GO2H=st143t3+406t2+1000t+1745+141st2+48s2+304st+128s+50s2t+7s3+s2t28t3+136t2+338t+308+6s3+26s2t+36st2+16s2+40st+8s+st265t3+213t2+203t+39+8s2t+52s+14s2+24st2+72st+2s3+st38t3+48t2+96t+64+s−59s3+15t3+9s2t+67s2+29st2+11t2−17s−11t+66st+s38t3+8s2+24st2+24s2p+s25t3+85s3+105st2+153s2t−51−s2−3t2−4st+9t+15s−2,(46)HGO1H=st783t42+1851t2+1114t3+1148st2+320s2t+9596t+126s2t2+354st3+441s2+10s4+1855st+136111+654s+12s3t+59s3+s2t24t4+734t2+192t3+4s4+28s3+16s2t2+56st2+32s2t+57s2+112st+1738t+28s+1421+st2185t42+629t3+1548t2+16st3+104st2+20s2t+4s2t2+25s2+216st+140s+1622t+13392+s38t4+8s4+24s2t2+8s2+8t2+48s3t+16st+48st3+16s3+24t3+48s2t+56st2+s236t4+109s4+68t3+40s3+224s3t+118st3+267s2t2+110s2t+191st2−s2+46t2+138st+10t+4s+72+s−108s4+t4−212s3t−10st3−109s2t2−20s3+4t3+48s2+3t2+126s2t+st2+10t−6s+23st−32−16t4+14s4+48s2t2+16s2+16t2+96s3t+72st+96st3+32s3+48t3+96s2t+112st2,HGO2H=st2t3+27t2+121t+1802+6t3+60t2+180t+1802+6t3+5st2+5s2+50t2+34st+126t+98+49t+s2t2+12t3+96t2+204t+1202−6t3+60t2+18t+1802−6t3+52t2+144t+1282+12t3+81t2+153t+842+5t2+65t+1802+20t2+130t+1802+5t2+25t+2502+6t3−5st2+s2t+48+42t2+28st+84t+4s2+32s2+3t2+33t+842+14400+30t3+90t2+3s2+3s2t+21st2+30+42st+40t+21s2−30t3+132t2+185t+842−16t3+2s2t+32s+72t2+4s2+48+104t+12st2+40st2+1216t3+96t2+192t+1282+7056+32400+4t250t+1802+16t2+96t+1282,PGOH=st12t3+27t2+121t+180+16t3+60t2+186t+180+112t3+96t2+204s+120−16t3+169t2+186t+180+15t2+65t+180−16t3+52t2+144t+128+16t3+5st2+5s2+50t2+34st+126t+98+49s+s2t+112t3+81t2+153t+84+16t3−5st2+s2t+48+42t2+28st+84t+4s2+32s+15t2+75t+250+1120+130t3+90t2+3s2+3s2t+21st2+30+42st+90t+121s(47)−130t3+132t2+186t+84+184−116t3+2s2t+32s+72s2+4s2+48+104t+112st2+40st+1216t3+96t2+192t+128+1180+14t2+50t+180+116t2+96t+128+130t3+2s3+9s+34t+9s2+58t2+17s2t+49st+42st2+13t2+33t+84−1216t3+2s3+24st2+16+48t2+12s2t+64st+48t+20s2+40s+112s3+6t3+37s2t−6t+23s2+29st2−3s+22st+2t2−2−16s3+16t3+26s2t+36st2+16s2+40st+24t2+8s+8t+13t3+44rt+16st2+2s2t+4s2+20+100t2+90t+24s+116t3+24st2+48t2+64st+8s2t+16s2+32t+32s−1254t3+162t2+16t+54+164t3+3s2+108t2+42st+27st2+12+66t+3s2t+15s+130t3+12s2t+12s2+18t+18s+48t2+142st2+60st+s2t16t3+60t2+186t+180+16t3+52t2+144t+128+16t3+16t3+26s2t+136st2+16s2+40st+24t2+8s+8t+st2130t3+132t2+186t+84−116t3+96t2+192t+128+116t3+2s2t+32s+12t2+4s2+48+104s+112st2+40st+1216t3+2s3+24st2+16+48t2+12s2t+64st+48t+20s2+40s+1254t3+162t2+162t+54+st31216t3+9s2+182t+128+s112s3+12t3+52s2t+8s2−8t2+52st2−4s−4t+116t3+16s3+48st2+48s2t(48)−130s3+6t3+2t+2s+34st2+58s2t−16s2−24st−8t2−16s3+30t3+16t2+34s2t+2t+2s+8t2+24st+52−1254s3+2t3+18st2+18s+54s2t−36st−54s2+6t−6t2−2−130s3+6t3+2t+2s+34st2+58s2t−16s2−24st−8t2+116t3+16s3+48st2+48s2t+158st2+3t3+6s3+34st2t+2s+2t+24st+8s2+16t2+s31216t3+16s3+48st2+48s2t+s2130s3+6t3+2t+2s+34st2+58s2t−16s2−24st−8t2−3216t3+16s3+48st2+48s2t+16s3+30t3+16t2+34s2t+2t+2s+8t2+24st+52st2+1254s3+2t3+18st2+18s+54s2t−36st−54s2+6t−6t2−2+130s3+6t3+2t+2s+34st2+58s2t−16s2−24st−8t2+116t3+16s3+48st2+48s2t−116t3+16s3+48st2+48s2t,SGOH=st1t2+11t+29+12t2+17t+29+12t2+15t+6s+st+21+13t2+22t+23(49)−12t2+17t+29−12t2+15t+24+13t2+20t+29+16t+35+16t+35+123+116t2+17t+9s+3st+1−16t2+23t+29−14t2+20t+24+119+129+15t+29+110t+29+16t2+2s2+13t+5s+6st+3+13s2+2t2+3t+9s+1−14t2+2s2+6st+8t+7s+2+16t2+2st+19s+15t+9+14t2+4st+10s+12t+4−129t2+24t+15+19t2+18t+4s+3st+7+16t2+6st+8s+11t+13+s2t12t2+17t29+12t2+15t+29+14t2+2s2+7s+8t+2+st216t2+23t+9+14t2+2st+5s+16t+14+129t2+24t+15+s13s+3t2+10st+2t+6s−1+14s2+4t2+8st+4s+4t−16s2+2t2+8st+3s+t−1−14s2+4t2+8st+4t+4s−129s2+t2+6st−1−16s2+2t2+8st+3s+t−1+16t2+8st+7t+5s+1+s3124s2+4t2+8st+4s+4t+s216s2+2t2+8st+t+3s−1−324s2+4t2+8st+4s+4t+14s2+4t2+8st+4s+4t+12s2+6t2+8st+7s+5t+129s2+6st+t2−1+16s2+2t2+8st+3s+t−1−14s2+4t2+8st+4s+4t,FH=st84t2+18s2+40st+263t+8s+476+s2t18t2+5s2+12st+58t+4s+77+s2t26t2+2s2+8st+52t+8s+26+st24t2+16+16t+s34t2+4s2+8st+s−17s2+7t2−6st+10t+14s−1+s220t2+36s2+40st−16s+4−8t2+8s2+16st.Proof.
For line distribution shown in Table2, use of formulas (5)–(9), (11)–(18), and (22) will give desired above expressions.Theorem 6.
LetH be the line graph of total of Kragujevac tree; K∈Kgq=s2t+1+1,s, then M-polynomial and certain topological indices that can be deducted from M-polynomial are(50)MH;u,v=stut+5vt+4+stu2t+4vt+5+stut+5v2t+s+2+stut+5v3t+3+sts−1ut+5v2t+4+sts−1ut+4v2t+4+stut+4v3t+3+stut+4v2t+s+2+stut+4v5+stut+4v3+stut+5v5+stu5v3+stu2t+s+2v3t+3+sts−1u3t+3v2t+4+sts−1u2t+s+2v2t+4+stt−1t−22u2t+4v2t+4+stu4v3+stu4v5+stu4vt+5+stu2t+4v4+stu2t+4v5+stu2t+s+2vs+3t+1+stt−12u2t+s+2v2t+s+2+stu3s+t−1v2t+s+2+su3s+t−1vs+3t−1+sts−1u2t+s+2v2s+t+ss−1s−22u2s+tv2s+t+ss−1u3s+t−1v2s+t+ss−1us+3t+1v2s+t+ss−12u3s+t−1v3s+t−1+ss−1u2t+2sv3s+t−1+s−1u2t+2sv2t+2s+stu2t+4vs+3t+1+stu2t+4v2t+2s+stt−12u3t+3v3t+3+stu3t+3vs+3t+1+stu3t+3v2t+2s+sus+3t+1v2t+2s+ss−1u2t+2sv2t+s,(51)M1H=stt+5+st2t+4+stt+5+stt+5+sts−1t+5+sts−1t+4+stt+4+stt+4+stt+5+st5+st2t+s+2+sts−13t+3+sts−12t+s+2+stt−1t−222t+4+st4+st4+st4+st2t+4+st2t+4+st2t+s+2+stt−122t+s+2+st3s+t−1+s3s+t−1+sts−12t+s+2+ss−1s−222s+2t+ss−13s+t−1+ss−1s+3t+1+ss−12t+2s+ss−123s+t−1+ss−12t+2s+s−12t+2s+st2t+4+st2t+4+stt−123t+3+st3t+3+st3t+3+ss+3t+1+stt+4+stt+5+st2t+s+2+st3t+3+sts−12t+4+sts−12t+4+st3t+3+st2t+s+2+stt+4+stt+4+st5+st3+st5+st3+st3t+3+sts−12t+4+sts−12t+4+stt−1t−222t+4+st3+st5+stt+5+st4+st5+sts+3t+1+stt−122t+s+2+st2t+s+2+ss+3t−1+sts−12t+2s+ss−1s−222t+2s+ss−12t+2s+ss−12t+2s+ss−12t+2s+ss−123s+t−1+ss−13s+t−1+s−12t+2s+sts+3t+1+st2t+2s+stt−123t+3+sts+3t+1+st2t+2s+s2t+2s,M2H=stt+4t+5+t+52t+4+2t+s+2t+5+3t+3t+5+s−12t+4t+5+s−1t+42t+4+3t+3t+4+2t+s+2t+4+5t+4+3t+4+5t+5+15+3t+32t+s+2s−12t+43t+3+s−12t+42t+s+2+t−1t−222t+42t+4+12+20+t+54+42t+4+52t+4+s+3t+12t+s+2+t−122t+s+22+3s+t−12t+s+2+s−12t+2s2t+s+2+s+3t+12t+4+2t+2s2t+4+t−123t+32+s+3t+13t+3+2t+2s3t+3+ss+3t−13s+t−1+s−12t+2s3s+t−1+s−12t+2ss+3t+1(52)+s−12t+2s2s+2t+s−123s+t−13s+t−1+s−12t+2s3s+t−1+s−1s−222s+2t2t+2s+s+3t+12s+2t+s−12t+2s2s+2t,M2mH=st1t+4t+5+1t+52t+4+12t+s+2t+5+13t+3t+5+s−112t+4t+5+s−112t+4t+4+13t+3t+4+12t+s+2t+4+15t+4+13t+4+15t+5+135+13t+32t+s+2+s−112t+43t+3+s−112t+42t+s+2+t−1t−2212t+42t+4+112+120+14t+5+142t+4+152t+4+1s+3t+12t+s+2+t−1212t+s+22t+s+2+12s+2t2t+4+12t+s+23s+t−1+s−112s+2t2t+s+2+1s+3t+12t+4+t−12(53)13t+33t+3+1s+3t+13t+3+12s+2t3t+3+ss−112s+2t3s+t−1+s−112s+2ts+3t+1+s−112s+2t2s+2t+s−1213s+t−13s+t−1+s−113s+t−12t+2s+1s+3t−1s+3t−1+s−1s−2212s+2t2s+2t+12s+2ts+3t+1+s−112s+2t2t+2s,RαH=st1t+4αt+5α+1t+5α2t+4α+12t+s+2αt+5α+13t+3αt+5α+s−112t+4αt+5α+s−112t+4αt+4α+13t+3αt+4α+12t+s+2αt+4α+15αt+4α+13αt+4α+15αt+5α+13α5α+13t+3α2t+s+2α+s−112t+4α3t+3α+s−112t+4α2t+s+2α+t−1t−2212t+4α2t+4α+13α4α+15α4α+1t+5α4α+14α2t+4α+15α2t+4α+1s+3t+1α2t+s+2α+t−1212t+s+2α2t+s+2α+12t+s+2α3s+t−1α+s−112s+2tα2t+s+2α+1s+3t+1α2t+4α+12s+2tα2t+4α+t−1213t+3α3t+3α+1s+3t+1α3t+3α+12s+2tα3t+3α+s12s+2tαs+3t+1α+s−112s+2tα3s+t−1α+s−112s+2tαs+3t+1α+s−112s+2tα2s+2tα+s−1213s+t−1α3s+t−1α+s−113s+t−1α2s+2tα+s−1s−2212s+2tα2s+2tα+1s+3t−1αs+3t−1α(54)+s−112s+2tα2s+2tα,RRαH=stt+4αt+5α+t+52t+4α+2t+s+2t+5α+3t+3αt+5α+s−12t+4αt+5α+s−12t+4αt+4α+3t+3αt+4α+2t+s+2α2t+s+2α+5αt+4α+3αt+4α+5αt+5α+3α5α+3t+3α2t+s+2α+s−12t+4α3t+3α+s−12t+4α2t+s+2α+t−1t−222t+4α2t+4α+3α4α+5α4α+t+5α4α+4α2t+4α+5α2t+4α+s+3t+1α2t+s+2α+t−122p+s+2α2t+s+2α+2t+s+2α3s+t−1α+s−12t+2sα2t+s+2α+s+3t+1α2t+4α+2t+2sα2t+4α+t−123t+3α3t+3α+s+3t+1α3t+3α+2t+2sα3t+3α+ss+3t−1α3s+t−1α+s−1s−222t+2sα2s+2tα+s−12t+2sα3s+t−1α+s−12t+2sαs+3t+1α+s−12t+2sα2s+2tα+s−123s+t−1α3s+t−1α+s−13s+t−1α2s+2tα+2t+2sαs+3t+1α+s−12t+2sα2s+2tα,SDDH=stt+5t+4+2t+4t+5+t+52t+s+2+t+53t+3+s−1t+52t+4+s−1t+42t+4+t+43t+3+t+42t+s+2+t+45+t+43+t+55+53+2t+s+23t+3+s−13t+32t+4+s−12t+s+22t+4+t−1t−22+43+45+4t+5+2t+44+2t+45+2t+s+2s+3t+1+t−122t+s+22t+s+2+3s+t−12t+s+2+s−12t+s+22s+2t+2t+4s+3t+1+2t+42s+2t+t−12+3t+3s+3t+1+3t+32s+2t+t+41t+5+t+512t+4+2t+s+21t+5+3t+31t+5+s−12t+41t+5+s+3t+112t+4+s−12t+41t+4+3t+31t+4+2t+s+21t+4+51t+4+31t+4+51t+5+315+3t+312t+s+2+s−12t+413t+3+s−12t+412t+s+2+t−1t−222t+412t+4+314+514+t+514+412t+4+512t+4+s+3t+112t+s+2(55)+t−122t+s+212t+s+2+2t+s+213s+t−1+s−12t+2s12t+s+212t+4+2t+2s12t+4+t−123t+313t+3+s+3t+113t+3+2t+2s13t+3+s3s+t−1s+3t−1+s−1s−222s+2t2s+2t+s−13s+t−12s+2t+s−1s+3t+12s+2t+s−12+s−12s+2t3s+t−1+s+3t+12s+2t+1+s−1s−22+s−12t+2s13s+t−1+s−12t+2s1s+3t+1+s−12t+2s12s+2t+s−12+s−13s+t−112s+2t+2t+2s1s+3t+1+s−12,HH=2st12t+9+13t+9+13t+s+7+14t+8+s−113t+9+s−113t+8+14t+7+13t+s+6+1t+9+1t+7+1t+10+18+15t+s+5+s−115t+7+s−114t+s+6+t−1t−2214t+8+17+19+1t+9+12t+8+12t+9+15t+2s+3+t−1214t+2s+4+14s+3t+1+s−114t+3s+2+1s+5t+5+12s+4t+4+t−1216t+6+1s+6t+4+15t+2s+3+2s14s+4t−2+s−1s−2214s+4t+s−115s+3t−1+s−113s+3t+1+s−114s+4t+s−1216s+2t−2+s−113t+5s−1+s13s+5t+1+2s−114s+4t,ISIH=stt+4t+52t+9+t+52t+43t+9+2t+s+22t+s+23t+s+7+3t+3t+54t+8+s−12t+4t+53t+9+s−12t+4t+43t+8+3t+3t+44t+7+2t+s+2t+43t+s+6+5t+4t+9+3t+4t+7+5t+5t+10+158+127+3t+32t+s+25t+s+5+s−12t+43t+35t+7+s−12t+42t+s+24t+s+6+t−1t−222t+42t+44t+8+209+t+54t+9+42t+42t+8+52t+42t+9+s+3t+12t+s+25t+2s+3+t−122t+s+22t+s+24t+2s+4+2t+s+23s+t−14s+3t+1+s−12t+2s2t+s+23t+3s+2+s+3t+12t+4s+5t+5+2t+2s2t+44t+2s+4+t−123t+33t+36t+6(56)+s+3t+13t+3s+6t+4+2t+2s3t+35t+2s+3+s2t+2ss+3t+13s+5t+1+s−1s−22s+2t2t+2s24s+4t+s−12t+2s3s+t−15s+3t−1+s−12t+2ss+3t+13s+5t+1+s−12t+2s2s+2t4s+4t+s−13s+t−13s+t−126s+2t−2+s−13s+t−12s+2t3t+5s+1+s+3t−13s+t−16s+4t−2+s−12t+2s2s+2t4s+4t,AH=stt+43t+532t+73+t+532t+433t+73+2t+s+23t+533t+s+53+3t+33t+534t+63+334353+s−12t+43t+533t+73+s−12t+43t+433t+63+3t+33t+434t+53+2t+s+232t+s+233t+s+43+53t+43t+73+33t+43t+53+53t+53t+83+335363+3t+332t+s+235t+s+33+532t+432t+73+s−12t+433t+335t+53+s−12t+432t+s+234t+s+43+2t+432t+43t−1t−284t+63+534373+t+5343t+73+432t+432t+63+2t+s+232t+s+23t−184t+2s+23+2t+2s32t+432s+4t+23+s+3t+132t+s+235t+2s+13+2t+s+233s+t−134s+3t−13+s−12t+2s32t+s+234t+3s3+s+3t+132t+43s+5t+33+3t+333t+33t−186t+43+s+3t+133t+33s+6t+23+2t+2s33t+332s+5t+13+s2t+2s3s+3t+133s+5t−13+2t+2s32s+2t3s−1s−284s+4t−23+s−12t+2s33s+t−135s+3t−53+s−12t+2s3s+3t+133s+5t−13+3s+t−133s+t−13s−186s+2t−43+s−13s+t−132s+2t33t+5s−33+s−12t+2s32s+2t34t+4s−23+s+3t−133s+t−134s+4t−23+s−12t+2s32s+2t34t+4s−23.Proof.
For line distribution shown in Table2, use of formulas given in (19)–(28) will give the above desired expressions.


## 4. Conclusion

In this study, we computed different topological indices and polynomials for total graph and the line graph of total graph of the considered tree,i.e., Kragujevac tree. It would be fascinating in the future to discover the same outcomes for those chemical invariants that have not yet been evaluated.



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*Source: 8695121-2021-11-18.xml*