# On Chemical Invariants of Semitotal-Point Graph and Its Line Structure of the Acyclic Kragujevac Network: A Novel Mathematical Analysis

Authors: Salma Kanwal; Rabia Safdar; Asia Rauf; Ammara Afzal; Wasim Jamshed; Meznah M. Alanazi; H. Y. Zahran
Journal: Journal of Chemistry
 (2022)

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## Abstract

Acyclic Kragujevac network is denoted byK; K∈Kgq=r2s+1+1,r. In this article, we have taken a deep look at some of the topological properties of the semitotal-point graph as well as its line structure by computing some algebraic polynomials. Few degree-dependent chemical invariants that can primarily be obtained via these algebraic expressions are also put forward.


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## Body

## 1. Introduction

A topological index, also known as a connectivity index, is a sort of molecular descriptor that is computed based on the chemical graph of a chemical compound in the fields of chemical graph theory, molecular topology, and mathematical chemistry. Topological indices are numerical factors that define the topology of a graph and are typically graph invariants. Topological indices are used in the progress of quantitative structure-activity relationships (QSARs), which correspond to biological activity or other characteristics of molecules with their chemical composition.The graphs under consideration in the following work are undirected, lack loops, as well as have bounded sets of nodes and edges. LetVL and EL indicate the node and line sets for graph L, respectively. Moreover, the cardinality of lines attached to a point denoted as dLu for u is its degree. Let R2,R3,R4,… be the rooted trees whose structure is shown in Figure 1. Another Kragujevac tree K from class Kg29,4 is shown in Figure 2.Figure 1
Rooted treesR2,R3,R4,R5.Figure 2
Acyclic Kragujevac structureK from family Kg29,4.A tree [1–7] consisting of a central vertex of degree not less than 3 and rooted trees of the form R1,R2,R3,… are joined to that tree is called a proper Kragujevac tree. Kgq,r denotes the class of proper Kragujevac trees K of order q and a central vertex with degree r. A tree derived by placing a new degree 2 node to a line linked to the degree one node of some proper acyclic Kragujevac network is termed as improper acyclic Kragujevac network, specified by Kgq,r∗.The rooted trees linked to the main vertex are specified asRt1,Rt2,…,Rtr where ti≥2;i=1,…,r. Here in this work, a unique case of the acyclic Kragujevac network is under consideration when Rt1,Rt2,...,Rtr are isomorphic, i.e., t1=t2=...=tr=s. Therefore, in this case, this structure K∈Kgq,r has a cardinality of nodes and lines q=r2s+1+1 and r2s+1, respectively.Topological indices are the mathematical calculations that are parallel to the forms of some simple finite graphs. The importance of topological indices is generally related with the quantitative structure property relationship (QSPR) and quantitative structure activity relationships (QSAR).Topological indices are chemical invariants that are derived from the degrees and distances between vertices. The Zagreb indices were put forward three decades before by Gutman and Trinajstić [8]. Later on, they were termed as Zagreb group indices by Balaban et al. [9, 10]. Then, the first Zagreb index is defined as follows:(1)M1L=∑t∈VLdt2=∑pt∈ELdp+dt.The second Zagreb index is defined as follows:(2)M2L=∑pt∈ELdpdt.The next member of this family of chemical invariants, denoted byM3L for a graph L, was brought to notice by Fath-Tabar [11] and is defined as follows:(3)M3L=∑pt∈ELdp−dt.The same authors put forward algebraic polynomials associated to the abovementioned chemical invariants, termed as Zagreb polynomials, given as follows:(4)M1L,u=∑pt∈ELudp+dt,M2L,u=∑pt∈ELudpdt,M3L,u=∑pt∈ELudp−dt.A modified version of the abovementioned invariants [12] were brought into existence. 1st and 2nd modified Zagreb indices are indicated, symbolized as mM1L and mM2L separately, and are computed as follows:(5)M1mL=∑pt∈EL1dp+dt.Shirdel et al. [13] reported a hyper invariant of Zagreb, defined as follows:(6)ℋℳL=∑pt∈ELdp+dt2.Ranjini et al. redefined the invariants of Zagreb [14] in 2013,expressing them as follows:(7)redefined first Zagreb index=ReM1L=∑pt∈ELdp+dtdpdt,redefined second Zagreb index=ReM2L=∑pt∈ELdpdtdp+dt,(8)redefined third Zagreb Index=ReM3L=∑pt∈ELdpdtdp+dt.Furtula et al. established the forgotten invariant, signified byℱL, in [15], which is stated as follows:(9)ℱL=∑t∈VLdt3=∑pt∈ELdp2+dt2.The forgotten algebraic expression deducing the forgotten invariant was also put forward, which is provided as follows:(10)FL,u=∑pt∈ELudp2+dt2.The following is a list of many other degree-dependent invariants by Gutman et al. [16] and V. R. Kulli [17–20]:(11)RRL=∑pt∈ELdpdt,RM2L=∑pt∈ELdp−1dt−1,and(12)RRRL=∑pt∈ELdp−1dt−1,first Gourava index=G1OL=∑pt∈ELdp+dt+dpdt,second Gourava index=G2OL=∑pt∈ELdp+dtdpdt,product connectivity Gourava index=PGOL=∑pt∈EL1dp+dtdpdt,sum connectivity Gourava index=SGOL=∑pt∈EL1dp+dtdpdt,first hyper−Gourava index=HGO1L=∑pt∈ELdp+dtdpdt2,(13)second hyper Gourava index=HGO2L=∑pt∈ELdp+dtdpdt2.M-polynomial [21] is given as follows:(14)ML;u,v=∑δ≤r≤s≤ΔxrsLurvs,where δ and Δ designate the min and max of vertex degrees individually, and xrs represents the number of edges pt∈EL such that dp,dt=r,s. The list of invariants deduced from previously defined algebraic expressions is as follows:(15)first Zagreb index=M1L=Da+DbML;a,ba=b=1,second Zagreb index=M2L=DaDbML;a,ba=b=1,modified second Zagreb index=M2mL=SaSbML;a,ba=b=1,Randic index′=RαL=SaαSbαML;a,ba=b=1=∑pt∈ELdpdtα,inverse Randic index′=RRL=DaαDbαML;a,ba=b=1=∑pt∈ELdpdt,and the symmetric division degree index is as follows:(16)SDDL=DaSb+DbSaML;a,ba=b=1=∑pt∈ELmindp,dtmaxdp,dt+maxdp,dtmindp,dt,harmonic index=HL=2SaLML;a,ba=b=1=∑pt∈EL2dp+dt,inverse sum indeg index=ISIL=SaLDaDbML;a,ba=b=1=∑pt∈ELdpdtdp+dt,and the augmented Zagreb index is as follows:(17)AL=Sa3Q−2Da3Db3ML;a,ba=b=1=∑pt∈ELdpdtdp+dt−23,where(18)DaML;a,b=a∂ML;a,b∂a,DbML;a,b=b∂ML;a,b∂b,SaML;a,b=∫0aML;t,btdt,SbML;a,b=∫0bML;a,ttdt,(19)LML;a,b=ML;a,a,QαML;a,b=aαML;a,b.Subdivided structure S(L) of some graph L is attained by inserting a vertex into each edge of the graph. Semitotal-point graph, i.e.,RL is derived from SL by linking each pair of connected white vertices, and semitotal-line graph, i.e., QL is derived from SL by linking each pair of connected black vertices. The semitotal-point graph and line graph of the semitotal-point graph of the Kragujevac tree are expressed in Figures 3 and individually in 7. The novelty of this work lies in the fact that topological invariants have become extensively important in chemical graph theory in determining chemical characteristics such as pressure and boiling point. Before the existence of topological invariants, these characteristics of molecular structures were being examined in the laboratory with wastage of money and time. Investigation of chemical invariants for graphic structures after applying some of unary or binary graphic operations gives rise to a new era of research for keen researchers.Figure 3
Semitotal-point graph for the Kragujevac network.Figure 4
The visual representation ofHML, ReM1L, ReM2L, ReM3L, M1mL, and RM2L.Figure 5
Graphic display forRRRL, GO1L, GO2L, HGO1L, HGO2L, and PGOL.Figure 6
The visual representation ofSGOL, FL, M1L, M2L, M2m, and SDDL.Figure 7
Line graph of semitotal-point graph of the Kragujevac tree.


## 2. Several Algebraic Polynomials and Invariants forRK

This portion of work provides evaluation of specified topological indices and polynomials as well as several degree-dependent invariants regarding to these polynomials, as described in this article for the semitotal-point graph of the Kragujevac tree. Figure3 depicts the semitotal-point graph of the Kragujevac network.(i)
ForL=ℛK, following are the forgotten and Zagreb polynomials:(20)FL,u=ru4r2+4+u4r2+4s2+8s+4+u4s2+8s+8+rsu4s2+8s+8+u4s2+8s+20+3u20+u8,M1L,u=ru2r+2+u2r+2s+2+u2s+4+rsu2s+4+u2s+6+3u6+u4,M2L,u=ru4r+u4rs+4r+u4s+4+rsu4s+4+u8s+8+3u8+u4,M3L,u=ru2r−2+u2r−2s−2+u2s+rsu2s+u2s−2+3u2+1.Take into account the Kragujevac tree’s semitotal-point graph, symbolized by ℛK. The edges of ℛK are classified based on degrees of end points Edk,dt and kt∈EℛK. Table 1 depicts classification of lines.(21)FL,u=∑pt∈ELudp2+dt2,=ru2r2+22+ru2r2+2s+22+rsu2s+22+22+ru2s+22+42+3rsu42+22+rsu22+22,=ru4r2+4+u4r2+4s2+8s+4+u4s2+8s+8+rsu4s2+8s+8+u4s2+8s+20+3u20+u8,M1L,u=∑pt∈ELudp+dt=ru2r+2+ru2r+2s+2+rsu2s+2+2+ru2s+2+2+rsu2s+2+4+3rsu4+2+rsu2+2=ru2r+2+u2r+2s+2+u2s+4+rsu2s+4+u2s+6+3u6+u4,M2L,u=∑pt∈ELudpdt=ru2r×2+ru2r×2s+2+rsu2s+2×2+ru2s+2×2+rsu2s+2×4+3rsu4×2+rsu2×2=ru4r+u4rs+4r+u4s+4+rsu4s+4+u8s+8+3u8+u4,M3L,u=∑pt∈ELudp−dt=ru2r−2+ru2r−2s−2+rsu2s+2−2+ru2s+2−2+rsu2s+2−4+3rsu4−2+rsu2−2=ru2r−2+u2r−2s−2+u2s+rsu2s+u2s−2+3u2+1.(ii)
ForL=ℛK, certain chemical invariants are given as follows:(22)HML=8r3+16r2+24r+200rs+8rs3+48rs2+8r2s,ReM1L=r+12+3r2s+2+s2s+2+12s+1+rs212s+2+14p+4+rs1s+1+34s+4+134+12s+2,ReM2L=r22r+1+2sr+s+1+2r+s+1+rs22s+2+4s+3+r2s+2+rs4s+2+4s+3+5,ReM3L=16r3+16r2+16r+248rs+8r3s+8r2s2+16r2s+24rs3+96rs2,M1Lm=r12r+2+12r+2s+2+12s+4+rs12s+4+12s+6+12+14,RM2L=4r2−r+8rs2+4r2s+14rs,(23)RRRL=r2r−1+4rs+2r−2s−1+2s+1+rs2s+1+6s+3+33+1,GO1L=12r2+12r+80rs+4r2s+16rs2,GO2L=16r3+16r2+16r+246rs+8r3s+8r2s2+16r2s+24rs3+94rs2,HGO1L=72r3+48r2+8r+1080rs+416rs2+40r2s+16r3s2+136rs3+48r3s+16r2s2,HGO2L=128r5+128r3+192r4+64rs5+448rs4+1216rs3+1699rs2+1123rs+256r+1024r3s5+6144r3s3+2560r3s4+2560r3s2+1024r3s+64r5s2+128r5s+128r3s3+208r3s+320r3s2+64r4s3+144r4s+192r4s2+7168rs,PGOL=r18r2+8r+18s2r+8rs2+8r2+16rs+8r+18s2+24s+16+rs18s2+24s+16+116s2+64s+48+348+116,SGOL=r16r+2+16r+4rs+2s+2+16s+8+rs16s+8+110s+14+314+18,FL=8r3+16r+112rs+8rs3+20rs2. For the line classifications depicted in Table 1, the use of formulas (6), (11)–(13), and (15) will produce the abovementioned results.(iii)
ForL=ℛK, chemical invariants deduced from M-polynomial are as follows:(24)ML;u,v=ru2rv2+ru2rv2s+2+rsu2s+2v2+ru2s+2v2+rsu2s+2v4+3rsu4v2+rsu2v2,M1L=4r2+4rs2+8r+36rs,M2L=8r2+4r2s+12rs2+48r+44rs,M2Lm=14+14s+1+rs4s+4+r4s+4+rs8s+8+3rs8+rs4,RαL=r2α2rα+r2s+2α2s+2α+rs2α2s+2α+r2α2s+2α+rs4α2s+2α+3rs2α4α+rs2α2α,RRαL=r2rα2α+r2s+2α2rα+rs2s+2α2α+r2s+2α2α+rs2s+2α4α+3rs4α2α+rs2α2α,SDDL=r21+1s+1+3rs22+r1+1s+1+rs3s+1+242+2+s,HL=r1r+1+1r+s+1+1s+2+rs1s+2+1s+3+32,ISIL=r22r+1+2sr+s+1+2r+s+1+s22rs+2+4rs+3+rs4s+2+4s+3+5+4r2s+4,AL=r2r223+r2s+232r312r+2s3+2rs23+r23+rs2s+234312s+43+3rs23.Table 1
Classification of lines ofℛK.
CardinalityEdge subfamilyrE2r,2rE2r,2s+2s+1rE2s+2,2rsE2s+2,43rsE4,2rsE2,2For the line classification depicted in Table1, the use of formulas (14)–(17) will produce the abovementioned results.Figures4, 5, 6, and 8 present the graphical display of certain topological indices and polynomials for the semitotal-point graph of the Kragujevac tree.Figure 8
The visual representation ofHL, ISIL, HML, ReM1L, ReM2L, and ReM3L.


## 3. Several Algebraic Polynomials and Invariants forLRK

This portion of work, provides evaluation of specified topological indices and polynomials as well as several degree-dependent invariants regarding to these polynomials, as described in this article for the semitotal-point graph of the Kragujevac tree. Figure7 depicts the line structure of a semitotal-point graph applied to K.(i)
ForL=LRK, following are the forgotten and Zagreb polynomials:(25)FL,u=r2u8s2+16rs+8r22+u4s2+8rs+8r2+u8r22+ru8s2+4+8s+4r2+8rs+u4s2+8s+4+4r2−u8r22−u8s2+8r2+16rs2+rsu8s2+4r2+8rs+4+8s+u8s2+4r2+8rs+16s+16+u8s2+16s+82−u8s2+32s+322+u4s2+8s+20+u4s2+16s+32+3u32+2u20+u8s2+24s+20+u8s2+16s+8+rs2u8s216s+82+2u8s2+24s+20+u8s2+32s+32,M1L,u=r2u4s+4r2+u2s+4r+u4r2+ru2s+2r+2+u4s+2r+2−u4s+4r2−u4r2+rsu4s+2r+2+u4s+2r+4−u4s+42−u4s+82+u2s+6+u2s+8+3u8+2u6+u4s+6+u4s+6+rs2u4s+42+u4s+6+u4s+8,M2L,u=r2u4s2+8rs+4r22+u4rs+4r2+u4r22+ru4rs+4r+u4s2+4rs+4s+4r−u4r22−u4s2+8rs+4r22+rsu4s2+4s+4rs+4r+u4s2+8s+4rs+8r−u4s2+8s+42−u4s2+16s+162+u8s+8+3u8s+16+3u16+2u8+u4s2+12s+8+u4s2+8s+4+rs2u4s2+2s+8+u4s2+8s+42+u4s2+16s+162,M3L,u=r2u2s+1+ru2s−2r+2+u2−2r−1+rsu2r−2+u2r−4+u2s−2+u2s+2u2+3+u−2+rs2u−2+1. Take into account the Kragujevac tree’s line graph of semitotal-point graph, symbolized by £ℛK. £ℛK consists of the nodes having degrees 2s+2r, 2r, 2s+4, 2s+2, 4, 2, and 2r. The lines of ℛK are classified based on degrees of their end nodes, Edk,dt and kt is a line in L. Table 2 depicts classification of lines.(26)FL,u=∑pt∈ELudp2+dl2=rr−12u2s+2r2+2s+2r2+r2u2s+2r2+2r2+rsu2s+2r2+2s+22+rsu2s+2r2+2s+42+rss−12u2s+22+2s+22+s2ru2s+22+2s+42+rss−12u2s+42+2s+42+rsu2s+22+42+rsu2s+42+42+3rsu42+42+2rsu42+22+rr−12u2r2+2r2+rsu2s+22+2s+42+ru2s+22+2r2+ru2s+22+2s+2r2+rsu2s+22+2s+22=r2u8s2+16rs+8r22+u4s2+8rs+8r2+u8r22+ru8s2+4+8s+4r2+8rs+u4s2+8s+4+4r2−u8r22−u8s2+8r2+16rs2+rsu8s2+4r2+8rs+4+8s+3u32+u8s2+4r2+8rs+16s+16+u8s2+16s+82−u8s2+32s+322+u4s2+8s+20+u4s2+16s+32+2u20+u8s2+24s+20+u8s2+16s+8+rs2u8s216s+82+2u8s2+24s+20+u8s2+32s+32.(27)M1L,u=∑pt∈ELudp+dt=rr−12u2s+2r+2s+2r+r2u2s+2r+2r+rsu2s+2r+2s+2+rsu2s+2r+2s+4+rss−12u2s+2+2s+2+s2ru2s+4+2s+4+rss−12u2s+4+2s+4+rsu2s+2+4+rsu2s+4+4+3rsu4+4+2rsu4+2+rr−12u2r+2r+rsu2s+2+2s+4+ru2s+2+2r+ru2s+2+2s+2r+rsu2s+2+2s+2=r2u4s+4r2+u2s+4r+u4r2+ru2s+2r+2+u4s+2r+2−u4s+4r2−u4r2+rsu4s+2r+2+u4s+2r+4−u4s+42−u4s+82+u2s+6+u2s+8+3u8+2u6+u4s+6+u4s+6+rs2u4s+42+u4s+6+u4s+8M2L,u=∑pt∈ELudpdt=rr−12u2s+2r×2s+2r+r2u2s+2r×2r+rsu2s+2r×2s+2+rsu2s+2r×2s+4+rss−12u2s+2×2s+2+s2ru2s+4×2s+4+rss−12u2s+4×2s+4+rsu2s+2×4+rsu2s+4×4+3rsu4×4+2rsu4×2+rr−12u2r×2r+rsu2s+2×2s+4+ru2s+2×2r+ru2s+2×2s+2r+rsu2s+2×2s+2=r2u4s2+8rs+4r22+u4rs+4r2+u4r22+ru4rs+4r+u4s2+4rs+4s+4r−u4r22−u4s2+8rs+4r22+rs3u8s+16+3u16+u4s2+4s+4rs+4r+u4s2+8s+4rs+8r−u4s2+8s+42−u4s2+16s+162+u8s+8+2u8+u4s2+12s+8+u4s2+8s+4+rs2u4s2+2s+8+u4s2+8s+42+u4s2+16s+162.M3L,u=∑pt∈ELudp−dt=rr−12u2s+2r−2s+2r+r2u2s−2r+2r+rsu2s+2r−2s+2+rsu2s+2r−2s+4+rss−12u2s+2−2s+2+s2ru2s+4−2s+4+rss−12u2s+4−2s+4+rsu2s+2−4+rsu2s+4−4+3rsu4−4+2rsu4−2+rr−12u2r−2r+rsu2s+2−2s+4+ru2s+2−2r+ru2s+2−2s+2r+rsu2s+2−2s+2=r2u2s+1+ru2s−2r+2+u2−2r−1+rsu2r−2+u2r−4+u2s−2+u2s+2u2+3+u−2+rs2u−2+1.(ii)
ForL=£ℛK, certain chemical invariants are given as follows:(28)HML=44r2s2+32r4+40r3s+224rs2−8r3+24r2s+152rs3+420rs+32rs4+8r+16r2,ReM1L=r312s2+4rs+2r2+r21r+s−12s2+4rs+2r2+r2s12s2+4rs+2r2+12s2+2s+2rs+2r+12s2+4s+2rs+4r+rs21s2+rs+s+r+1s2+2s+rs+2r+12s2+4s+2+32s2+6s+4−12s2+8s+8+14s+4+14s+8+44s2+8s+4+1s2+3s+2+1s2+2s+4+rs1s2+4s+4−12s2+8s+8+14s+4+14s+8+44s2+8s+4+1s2+3s+2+1s2+2s+4+rs1s2+4s+4−24s2+8s+4−12s2+4rs+2r2+1s2+2s+rs+2r+1s2+2s+1+12s+2r−1s2+2s+rs+2r−1s2+2s+4+3+64s2+12s+8+34s+4+22s+4+1s2+rs+s+r+14s2+8s+4+r2−12+rs31s2+3s+8+12s2+8s+8+1s2+3s+2+s12s+2+12s+2+r2s+2+12s2+2rs+2s+2r,(29)ReM2L=r2s212s+2r+22s+r+1+22s+r+2+r3s1r+s+22r+s+r412s+2r+2r+s+rs222s+r+1−12r+2s+42s+r+2+12s+2−1s+1+42s+3+2s+2−2s+2+4s+3+4s+4+62s+3+22s+r+1+2s+1+r312−12s+2r+r3s2s+2r+sr222s+r+1−1r+s+42s+r+2+2r+s+1+22s+r+1+rs322s+r+1+22s+r+2+1s+1−12s+2+62s+3+2s+2−12s+4+22s+3+1s+1+rs412s+2+22s+3+12s+4+rs4s+3−12s+2−2s+2+8s+4+488+83+42s+3+2r+2s+1+1s+1+r2−12+r22r+s+1,(30)ReM3L=48r2s3+56r3s2+48r4s+32r5+376rs3−16r4+16r3s+96r2s2+192rs4+360rs2+72r2s+656rs+16r3+8r2,M1mL=r218s+8r+12s+4r+r12s+2r+2+14s+2r+2−18s+8r+18+rs14s+2r+2+14s+2r+4−18s+8−14s+8+12s+6+12s+8+38+26+14s+6+14s+4+rs218s+8+14s+6+18s+16−18+1r8RM2L=10r2s2+8r3s+8r2s+8r4−12r3+10r2+30rs2+40rs+28rs3+8rs4−3,RRRL=r24s2+4r2+8rs−4s−4r+12+4r2+4rs−4r−2s+1+4r2−4r+12+r4rs+2r−2s−1+4s2+4rs−1−4s2+4r2+8rs−4s−4r+12−4r2−4r+12+rs4s2+4rs++2r−1+4s2+4rs+6r+4s−3−4s2+12s+92+6s+3+4s2+8s+3−4s2+4s+12+4s2+4s+1+6s+9+9+23+rs24s2+4s+12+4s2+8s+3+4s2+12s+92.(31)GO1L=10r2s2+8r3s+8r4+8r3+24r2s+84rs2+8r2+158rs+44rs3+8rs4+4r,GO2L=48r2s2+56r3s2+48r4s+32r5+376rs3−16r4+16r3s+96r2s2+192rs4+360rs2+72r2s+656rs+16r3+8r2,HGO1L=8r2s4+32r6+48r5+100r3s+244r3s3+64r4s2+80r3s2+264r2s3+64r4s+32rs6+204r2s2+8r2s+920rs4−16r3+1224rs2+288rs5+32r5s+144r2s4+1288rs3+2784rs,HGO2L=896r2s6+512r8+1536r7s+1664r3s5+1920r4s5+2008r5s3+883r6s2+20544rs6−256r7−768r6s+21760r2s5+1088r3r4−1536r4s3+256r5s−691r5s2+82r6s2+448r4s4+4864rs7+128r3s6+37876rs5+6208r2s4+960r3s+1152r4s+42880rs3+128r2s2+1920r4s2+1244r3s2+4928r2s3+47988rs4+512rs8+4352r3s3+17184rs2+55622rs+128r5+768r2s5+64r3+128r4+356rs7+128r2s+384r4s3+896r2s2+64r5s2,(32)PGOL=r2216s3+16r3+48r2s+48rs2−r216s3+16r3+48r2s+48rs2+r28rs224r2s+16r3+rs16s3+24s2+24rs2+32rs+8r2s+8r2+8s+8r+rs16s3+40s2+24rs2+56rs+8r2s+16r2+16s+16r+rs16s2+96s+128−r216s3+48s2+48s+16+rs16s2+64s+48+rs2216s3+48s2+48s+16+rs216s3+72s2+104p+48+rs2216s3+96s2+192s+128+3rs128+2rs48+r2216r3−r216r3−rs216s3+96s2+192s+128+rs16s3+72s2+104s+48+r16s3+24rs2+24s2+32rs+8r2s+8r2+8s+8r+rs16s3+48s2+48s+16,(33)SGOL=rr−1214s2+8rs+4r2+4r+4s+r214rs+4r2+2s+4r+rp14s2+4s+4rs+4r+4s+2r+2+rs14s2+8s+4rs+8r+4s+2r+2+rss−1214s2+8s+4+4s+4+rs214s2+12s+8+4s+6+rss−1214s2+16s+16+4s+8+rs18s+8+2s+6+rs18s+16+2s+8+3rs116+8+2rs18+6+rr−1214r2+4r+rs14s2+12s+8+4s+6+r14rs+4r+2r+2s+2FL=24r2s2+16r4+4r3s+128rs2+88rs3+304rs+14rs4+8r. For edge subfamilies and their cardinalities represented in Table 2, application of expressions (6)–(14) and (15) will produce the abovementioned results.(iii)
ForL=£ℛK, chemical invariants deduced from M-polynomial are as follows:(34)ML;u,v=rr−12u2s+2rv2s+2r+r2u2s+2rv2r+rsu2s+2rv2s+2+rsu2s+2rv2s+4+rsu2s+4v4+rss−12u2s+2v2s+2+rs2u2s+2v2s+4+rss−12u2s+4v2s+4+rsu2s+2v4+3rsu4v4+2rsu4v2+rr−12u2rv2r+rsu2s+2v2s+4+ru2s+2v2r+ru2s+2v2s+2r+rsu2s+2v2s+2,M1L=rr−122s+2r+r22s+2r+rs2s+2r+rs2s+2r+rs2s+2+r2s+r−rs+r+2s2+2r2s+2+2r3+rs2s+1−rss+1+r2s+2r+rs2r+2+rss−122s+2+rs22s+2+24rs+rs2s+2+3rs2s+2−rss+2+3rs2s+4+12rs+8rs+2r3−2r2+rs22s+4+rs2s+2−rss+2,M2L=8r3s+8r4+10r2s2+60rs2+16r2s−4r3+6r4+8rs4+94rs+8r2+36rs3,M2mL=r218s2+16rs+8r2+r14r+4s−18s2+16rs+8r2+14s2+4rs+4s+4r+rs14s2+4rs+4s+4r+14s2+8s+4rs+8r−18s2+16s+8−18s2+32s+32+18s+8+18s+16+316+14+14s2+12s+8+14s2+8s+4+rs218s2+16s+8+14s2+12s+8+18s2+32s+32+18−18r+14s+4,(35)RαL=r222s+2rα2s+2rα−r22s+2rα2s+2rα+r22rα2s+2rα+rs2s+2α2s+2rα+rs2s+4α2s+2rα+rs222s+2α2s+2α−rs22s+2α2s+2α+rs22s+4α2s+2α+rs222s+4α2s+4α−rs22s+4α2s+4α+rs4αs+2α+rs4α2s+4α+3rs4α4α+2rs2α4α+r222rα2rα−r22rα2rα+rs2s+4α2s+2α+r2rα2s+2α+r2s+2rα2s+2α+rs2s+2α2s+2α,(36)RRαL=rr−122s+2rα2s+2rα+r22rα2s+2rα+rs2s+2α2s+2rα+rs2s+4α2s+2rα+rss−122s+2α2s+2α+rs22s+4α2s+2α+rss−122s+4α2s+4α+rs4α2s+2α+rs4α2s+4α+3rs4α4α+2rs2α4α+rr−122rα2rα+rs2s+4α2s+2α+r2rα2s+2α+r2s+2rα2s+2α+rs2s+2α2s+2α,(37)SDDL=r252−12s+2r+2s+1+r1s+r−32+rs272+4s+r−12s+2r−12s+2+2s+2+6s+1+rs25s+1+5s+2+2s+r−12s+2−12s+4+2+r2s1s+1+1s+2+12s+2r+rs31s+2+1s+1+12s+2+12s+4+s+1,(38)HL=r214s+4sr+1s+2r+r12s+r+1+1s+r+1+14−14s+4r+rs12s+r+2+12s+r+1−14s+4−14s+8+1s+3+1s+4+34+23+12s+3+12s+2+rs214s+4+12s+3+14s+8−14,ISIL=rr−122s+2r2s+2r4s+4r+r22r2s+2r2s+4r+rs2s+22s+2r4s+2r+2+rs2s+42s+2r4s+2r+4+rss−122s+22s+24s+4+rs22s+42s+24s+6+rss−122s+42s+44s+8+rs42s+22s+6+rs42s+42s+8+3rs448+2rs246+rr−122r2r4r+rs2s+42s+24s+6+r2r2s+22s+2r+2+r2s+2r2s+24s+2r+2+rs2s+22s+24s+4,(39)AL=rr−122s+2r32s+2r314s+4r−23+r22r32s+2r312s+4r−23+rs2s+232s+2r314s+2r3+rs2s+432s+2r314s+2r+23+rss−122s+232s+2314s+23+rs22s+432s+2314s+43+3rs434316+rs432s+2312s+43+3rs432s+4312s+63+rss−122s+432s+4314s+63+2rs2343143+rr−122r32r314r−23+rs2s+432s+2314s+43++r2r32s+2312s+2r3+r2s+2r32s+2314s+2r3+rs2s+232s+2314s+2.Table 2
Line categorization ofLRK.
CardinalityEdge subfamilyrr−1/2E2s+2r,2s+2rr2E2s+2r,2rrsE2s+2r,2s+2rsE2s+2r,2s+4rss−1/2E2s+2,2s+2s2rE2s+2,2s+4rss−1/2E2s+4,2s+4rsE2s+2,43rsE4,42rsE4,2rr−1/2E2r,2rrsE2s+2,2s+4rE2s+2,2rrE2s+2,2s+2rrsE2s+2,4Edge subfamilies are presented in Table2, with their cardinalities, and the use of formulas given in (14)–(17) will produce the abovementioned results. A space visual display of various chemical indices is presented in Figures 9, 10, and 11 for the line structure of semitotal-point graph of K.Figure 9
3-D visual representation ofM1Lm, RM2L, RRRL, GO1L, GO2L, and HGO1L.Figure 10
The visual representation ofHGO2L, PGOL, SGOL, FL, M1L, and M2L.Figure 11
The visual representation ofM2mL, SDDL, HL, and ISIL.


## 4. Conclusion

The estimation of topological properties is critical in chemical graph theory. Semitotal-point operations and the line structure of semitotal-point graphs of acyclic Kragujevac networks are addressed in this study. We quantified certain polynomials, chemical invariants, and further invariants that can be derived from computed polynomials. Vertex degrees that play a vital role in chemical graph theory were under consideration throughout this work. We investigated some families of invariants like Zagreb, hyper Zagreb, Gourava invariants, and many more.



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*Source: 7995704-2022-02-22.xml*