# Computing FGZ Index of Sum Graphs under Strong Product

Authors: Zhi-Ba Peng; Saira Javed; Muhammad Javaid; Jia-Bao Liu
Journal: Journal of Mathematics
 (2021)

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## Abstract

Topological index (TI) is a function that assigns a numeric value to a (molecular) graph that predicts its various physical and structural properties. In this paper, we study the sum graphs (S-sum, R-sum, Q-sum and T-sum) using the subdivision related operations and strong product of graphs which create hexagonal chains isomorphic to many chemical compounds. Mainly, the exact values of first general Zagreb index (FGZI) for four sum graphs are obtained. At the end, FGZI of the two particular families of sum graphs are also computed as applications of the main results. Moreover, the dominating role of the FGZI among these sum graphs is also shown using the numerical values and their graphical presentations.


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## Body

## 1. Introduction

LetG=VG,EG be a (molecular) graph with VG and EG as sets of vertex and edge respectively. The degree of a vertex vεVG is the number of edges which are incident to v. A topological index (TI) is a function that assigns a numeric value to the under study (molecular) graph, see [1–3]. TIs are used to predict the various chemical and structural properties such as octane isomers including entropy, acentric factor, density, total surface area, molar volume, boiling point, capacity of heat at temperature and pressure, enthalpy of formation, connectivity of compounds and octanol water partition, see [4]. These are also used to study the quantitative structure property and activity relationships which are very important in the subject of cheminformatics, see [5].Wiener see [6] defined a distance-based TI to compute the boiling point of paraffin. Gutman and Trinajstic see [7] investigated total π-electron energy of the molecule graphs using a degree-based TI called by first Zagreb index nowadays. Later on, various Zagreb indices (hyper, forgotten, multiplicative and augmented) are defined to predict the different physicochemical properties chirality, heterosystems, complexity, branching and ZE-isomerism of the (molecular) graphs see [8]. Li and Zheng defined the first general Zagreb index (FGZI) and studied its different properties see [9].The operations of graphs (addition, deletion, complement, product, union and intersection) also play a very important role for the construction of new graphs, see [10]. In particular, for a graph G Yan et al. [11] introduced the five new graphs SG, RG, LG, QG and TG are defined with the help of the operations of subdivision S, semitotal point R, Line graph L, semitotal edge Q and total vertex and edge T respectively. Also, Wiener index of these ϕG graphs is computed, where Φ∈S,R,L,Q,T. Eliasi and Taeri [12] constructed the Φ-sum graphs GΦ+H by the Cartesian product of the graphs ΦG and H, where ΦG is obtained after applying the Φ on G for Φ∈S,R,Q,T. Moreover, they computed the Wiener indices of the Φ-sums graphs GS+H, GR+H, GQ+H and GT+H. Deng et al. [13] computed the first and second Zagreb indices of four operations on graphs using the concept of Cartesian product. Akhter and Imran [14] computed the forgotten topological index of four operations on graphs under Cartesian product. Liu et al. [15] computed the FGZI of Φ-sum graphs under the operation of Cartesian product. Sarala et al. [16] computed the F-index of Φ-sum graphs under the operation of strong product. For further study, we refer to [17–29].In this paper, we extend this study and compute FGZI of theΦ-sums graphs (GΦ⊠H) under the operation of strong product of ΦG and H in term of FGZI of its factor graphs G and H, where Φ∈S,R,Q,T, G and H are any two connected graphs. The obtained results are general extension of the results of Deng et al. [13], Akhter and Imran [14], Liu et al. [15] and Sarala et al. [16] works. Forthcoming sections are arranged as: Section 2 includes the basic formulae and results, Section 3 covers the main results and Section 4 includes the conclusion.


## 2. Preliminaries

An atom is presented by a vertex and bonding between atom is showed by edge in the molecular graphs. The first and second Zagreb indicesM1G and M2G of G are defined as follows [5, 7, 22]:(1)M1G=∑v∈VGdGv2=∑uv∈EGdGu+dGv,M2G=∑uv∈EGdGudGv.For any real numberα FGZ index and general Randic´ index are defined as(2)Mα=∑v∈VGdGαv=∑uv∈EGdGα−1u+dGα−1v,RαG=∑uv∈EGdGudGvα.For more details of aforesaid TIs, see [30, 31]. The following graphs are defined with the help of the subdivision related operations (S, R, Q and T), see [16].(i)
By inserting a new vertex in eachuv∈EG, S (G) is obtained.(ii)
By joining the end (original) vertices of the edges being incidence on each new vertex inS (G), R (G) is obtained.(iii)
By joining the new vertices which have common adjacent (original) vertices inS (G), Q (G) is obtained.(iv)
By operating bothR(G) and Q(G) on S (G), T (G) is obtained. For more explanation, see Figure 1.Figure 1
(a)G≅P4, (b) SG, (c) RG, (d) QG, and (e) TG.
(a)(b)(c)(d)(e)In 1960 Sabidussi introduced the strong product for any two graphsG∘H has vertex set Cartesian product VG×VH such that s1,t1 and s2,t2 will be adjacent in G∘H iff: [s1=s2 and t1 is adjacent to t2] or [t1=t2 and s1 is adjacent to s2] or [s1 is adjacent to s2 and t1 is adjacent to t2]. Now, the Φ-sum graph under the operation of strong product is defined in [16].LetG and H be two graphs. For Φ∈ {S, R, Q, T}, ΦG is a graph constructed by the operation Φ on G with set of vertex VΦG and set of edge EΦG. Then, Φ-sum graph GΦ⊠H under the strong product of graphs ΦG and H is a graph with vertex set(3)VGΦ⊠H=VΦG×VH=VG∪EG×VH.such that two vertex s1,t1 and s2,t2 of VGΦ⊠H are adjacent iff, either [s1=s2∈VG and t1t2∈EH] or [t1=t2∈VH and s1s2∈EΦG] or [s1s2∈EΦG and t1t2∈EH].It is observed thatGΦ⊠H has VH copies of ΦG which are labeled by the vertices of H. Also, VG and EG are shown as blue and red vertices in GΦ⊠H respectively, see Figure 2.Figure 2
ForG≅P3 and H≅P2, we have (a) P3S⊠P2, (b) P3R⊠P2, (c) P3Q⊠P2, and (d) P3T⊠P2.


## 3. Main Results

Now, we compute the analytical closed expression for FGZ index of theGS⊠H, GR⊠H, GQ⊠H and GT⊠H.Theorem 1.
LetG and H be two graphs. For α∈N+ and n=α−1, FGZ index of S-sum graph is(4)MαGS⊠H=∑m=0n∑k=0mnmmkM1n−m+kGM1m+1H+M1α−m+kGM1mH+M1α−m+kGM1m+1H+2n+1eG∑m=0nnmM1mH1+M1H.Proof.
By the definition for any vertexu,v∈VGS⊠H, the degree du,v of u,v is(5)du,v=du+dv+dudvifu∈VGdu+dudvifu∈VSG−VGdu,v=du+dv+dudvifu∈VG2+2dvifu∈VSG−VG,MαGS⊠H=∑u,v∈VGS⊠HdGS⊠Hαu,v=∑s1,t1s2,t2∈EGS⊠Hdns1,t1+dns2,t2=∑s1=s2∈VG∑t1t2∈EHdns1,t1+dns2,t2+∑t1=t2∈VH∑s1s2∈ESGdns1,t1+dns2,t2+∑s1s2∈ESG∑t1t2∈EHdns1,t1+dns2,t2=∑1+∑2+∑3.
For each vertexs1=s2∈VG and edge t1t2∈EH, we have(6)∑1=∑s1=s2∈VG∑t1t2∈EHdns1,t1+dns2,t2=∑s∈VG∑t1t2∈EHds+dt1+dsdt1n+ds+dt2+dsdt2n=∑s∈VG∑t1t2∈EH∑m=0n∑k=0mnmmkdn−msdm−kt1dksdkt1+dn−msdm−kt2dksdkt2=∑s∈VG∑t1t2∈EH∑m=0n∑k=0mnmmkdn−m+ksdkt1+dn−m+ksdmt2=∑s∈VG∑t1t2∈EH∑m=0n∑k=0mnmmkdn−m+ksdmt1+dmt2,∑1=∑m=0n∑k=0mnmmkM1n−m+kGM1m+1H.
We know thatESG=2EG. Also, for every t1=t2∈VH and s1s2∈ESG with s1∈VG, s2∈VSG−VG then,(7)∑2=∑t1=t2∈VH∑s1s2∈ESGdns1,t1+dns2,t2=∑t∈VH∑s1s2∈ESGds1+dt+ds1dtn+ds2+ds2dtn=∑t∈VH∑s1s2∈ESG∑m=0n∑k=0mnmmkdn−ms1dm−ktdks1dkt+∑m=0nnmdn−ms2dms2dmt=∑t∈VH∑s1s2∈ESG∑m=0nnm∑k=0mmkdn−m+ks1dmt+2n−m2dtm=∑t∈VH∑s1s2∈ESG∑m=0nnm∑k=0mmkdn−m+ks1dmt+2n−m+mdmt=∑m=0n∑k=0mnmmkM1n−m+k+1GM1mH+∑m=0nnm2n+1eGM1mH=∑m=0n∑k=0mnmmkM1α−m+kGM1mH+eG∑m=0nnm2n+1M1mH.∑3=∑s1s2∈ESG∑t1t2∈EHdns1,t1+dns2,t2=∑s1∈VG,s2∈VSG−VG∑t1t2∈EHdns1,t1+dns2,t2=∑s1∈VG,s2∈VSG−VG∑t1t2∈EHds1+dt1+ds1dt1n+ds2+ds2dt2n=∑s1∈VG,s2∈VSG−VG∑t1t2∈EH∑m=0n∑k=0mnmmkdn−ms1dm−kt1dks1dkt1+∑m=0nnmdn−ms2dms2dmt2=∑s1∈VG,s2∈VSG−VG∑t1t2∈EH∑m=0nnm∑k=0mmkdn−m+ks1dm−k+kt1+2n−m2mdmt2=∑m=0nn∑k=0mmk∑s1∈VG,s2∈VSG−VGdn−m+ks1∑t1t2∈EHdmt1+2n2eG∑t1t2∈EHdmt2m=∑m=0n∑k=0mnmmkM1α−m+kGM1m+1H+eG∑m=0nnm2n+1M1m+1H.
Hence(8)MαGS⊠H=∑m=0n∑k=0mnmmkM1n−m+kGM1m+1H+M1α−m+kGM1mH+M1α−m+kGM1m+1H+2n+1eG∑m=0nnmM1mH1+M1H.Theorem 2.
LetG and H be two graphs. For α∈N+ and n=α−1, FGZ index of R-sum graph is(9)MαGR⊠H=∑m=0n∑k=0mnmmk2n−m+kM1mHM1n−m+kGM1H+M1α−m+kG2+M1H+2∑m=0nnm2n+1eGM1m+1H.Proof.
By definition(10)MαGR⊠H=∑u,v∈VGR⊠Hdαu,v=∑s1,t1s2,t2∈EGR⊠Hdns1,t1+dns2,t2=∑s1=s2∈VG∑t1t2∈EHdns1,t1+dns2,t2+∑t1=t2∈VH∑s1s2∈ESGdns1,t1+dns2,t2+∑s1s2∈ESG∑t1t2∈EHdns1,t1+dns2,t2=∑1+∑2+∑3.
Consider,(11)∑1=∑s1=s2∈VG∑t1t2∈EHdns1,t1+dns2,t2=∑m=0n∑k=0mnmmk2n−m+kM1n−m+kGM1m+1H.
SincedRGs=2dGs For each vertex t1=t2∈EH and edge s1s2∈ERG where s1,s2∈VG then,(12)∑2=∑t1=t2∈VH∑s1s2∈ERGdns1,t1+dns2,t2=∑t1=t2∈VH∑s1s2∈ERG,s1,s2∈VGdns1,t1+dns2,t2+∑t1=t2∈VH∑s1s2∈ERG,s1∈VG,s2∈VRG−VGdns1,t1+dns2,t2=∑2′+∑2″.∑2′=∑t1=t2∈VH∑s1s2∈ERGdns1,t1+dns2,t2=∑t∈VH∑s1s2∈ERGds1+dt+ds1dtn+ds2+dt+ds2dtn=∑t∈VH∑s1s2∈ERG∑m=0n∑k=0mnmmkdn−ms1dm−ktds1dtk+dn−ms2dm−ktds2dtk=∑t∈VH∑s1s2∈EGs1,s2∈VG∑m=0n∑k=0mnmmk2n−m+kdn−m+ks1dmt+dn−m+ks2dmt=∑m=0n∑k=0mnmmk2n−m+kM1α−m+kGM1mH.
For eacht1=t2∈EH and s1s2∈ERG where s1∈VG ,s2∈VRG−VG then,(13)∑2″=∑t1=t2∈VH∑s1s2∈ERGdns1,t1+dns2,t2=∑t∈EH∑s1s2∈ERGds1+dt+ds1tn+ds2+ds2dtn=∑t∈VH∑s1s2∈ERG∑m=0nnm∑k=0mmkdn−ms1dm−ktds1dtk+dn−ms2ds2dtm=∑t∈VH∑s1∈VG∑m=0nnm∑k=0mmk2n−m+kdn−m+ks1dmt+2n−m2mdmt=∑t∈VH∑s1∈VG∑m=0nnm∑k=0mmk2n−m+kdn−m+ks1dmt+2ndmt=∑m=0nnm∑k=0mmk2n−m+k∑t∈VHdmt∑s1∈VGdn−m+ks1+2n∑t∈EHdmt2eG=∑m=0n∑k=0mnmmk2n−m+kM1mHM1α−m+kG+2n+1eG∑m=0nnmM1m+1H.
For eacht1t2∈EH and s1s2∈ERG, where s1,s2∈VG then,(14)∑3=∑s1s2∈ERG∑t1t2∈EHdns1,t1+dns2,t2∑3=∑s1s2∈ERGs1,s2∈VG∑t1t2∈EHdns1,t1+dns2,t2+∑s1s2∈ERG,s1∈VGs2∈VRG−VG∑t1t2∈EHdns1,t1+dns2,t2=∑t1t2∈EH∑s1,s2∈VG∑m=0n∑k=0mnmmk2ds1n−m+kdmt1+2ds2n−m+kdmt2=∑t1t2∈EH∑s1,s2∈VG∑m=0n∑k=0mnmmk2n−m+kdn−m+ks1dmt1+dn−m+ks2dmt2=∑3′+∑3″,∑3′=∑t1t2∈EH∑s1s2∈ERGdns1,t1+dns2,t2=∑t1t2∈EH∑s1s2∈ERGds1+dt1+ds1dt1n+ds2+dt2+ds2dt2n=∑t1t2∈EH∑s1s2∈ERG∑m=0n∑k=0mnmmkdn−m+ks1dmt1+dn−m+ks2dmt2=∑m=0n∑k=0mnmmk2n−m+kM1α−m+kGM1m+1H.
For eacht1t2∈EH and s1s2∈ERG, where s1∈VG,s2∈VRG−VG then,(15)∑3′′=∑t1t2∈EH∑s1s2∈ERGdns1,t1+dns2,t2=∑t1t2∈EH∑s1s2∈ERGds1+dt1+ds1dt1n+2+2dt2n=∑t1t2∈EH∑s1s2∈ERG∑m=0nnm∑k=0mmkdn−ms1dm−kt1ds1dt1k+2n−m2dt2m=∑t1t2∈EH∑s1,s2∈VG∑m=0nnm∑k=0mmk2ds1n−m+kdmt1+2n−m2dt2m=∑t1t2∈EH∑s1,s2∈VG∑m=0nnm∑k=0mmk2n−m+kdn−m+ks1dmt1+2ndmt2=∑m=0nnm∑k=0mmk2n−m+k∑s1,s2∈VGdGn−m+ks1∑t1t2∈EHdmt1+2n∑t1t2∈EHdmt22eG=∑m=0nnm∑k=0mmk2n−m+kM1α−m+kGM1m+1H+2n+1eGM1m+1H∑3′′=∑m=0nnm∑k=0mmk2n−m+kM1α−m+kGM1m+1H+2n+1eGM1m+1H.
Hence(16)MαGR⊠H=∑m=0n∑k=0mnmmk2n−m+kM1mHM1n−m+kGM1H+M1α−m+kG2+M1H+2∑m=0nnm2n+1eGM1m+1H.Theorem 3.
LetG and H be two graphs. For α∈N+ and n=α−1, FGZ index of Q-sum graph is(17)MαGQ⊠H=∑m=0n∑k=0mnmmkM1n+k−mGM1mHM1H+M1G+M1GM1H+∑m=0n∑i=0nnmni∑uv∈EGM1mHdGn−iudGiv2+2M1H+∑m=0nnm∑i=0nni∑uv,vw∈EGM1mHdGn−iudGiv+dGn−ivdGiw1+M1H.Proof.
By definition(18)MαGQ⊠H=∑u,v∈VGQ⊠Hdαu,v=∑s1,t1s2,t2∈EGQ⊠Hdns1,t1+dns2,t2=∑s1=s2∈VG∑t1t2∈EHdns1,t1+dns2,t2+∑t1t2∈EH∑s1s2∈EQGdns1,t1+dns2,t2+∑s1s2∈EQG∑t1t2∈EHdns1,t1+dns2,t2=∑1+∑2+∑3.
Consider,(19)∑1=∑s1=s2∈VG∑t1t2∈EHdns1,t1+dns2,t2=∑m=0n∑k=0mnmmkM1n−m+kGM1m+1H.
For each vertext1=t2∈VH and edge s1s2∈EQG where s1∈VG, s2∈VQG−VG. Note that dQGs2  = dGu+dGv for s2∈VQG−VG], s2 is the vertex inserted into the edge uv of G for all u, v∈VG, we have(20)∑2=∑t1=t2∈VH∑s1s2∈EQGdns1,t1+dns2,t2∑2=∑t1=t2∈VH∑s1s2∈EQG,s1∈VGs2∈VQG−VGdns1,t1+dns2,t2+∑t1=t2∈VH∑s1s2∈EQG,s1,s2∈VQG−VGdns1,t1+dns2,t2=∑2′+∑2′′,∑2′=∑t1=t2∈VH∑s1s2∈EQGdns1,t1+dns2,t2=∑t∈VH∑s1s2∈EQGds1+dt+ds1dtn+ds2+ds2dtn=∑t∈VH∑s1s2∈EQG∑m=0nnm∑k=0mmkdn−ms1dm−ktds1dtk+dn−ms2ds2dtm=∑t∈VH∑s1s2∈EQG∑m=0nnm∑k=0mmkdn−ms1dm−ktds1dtk+dns2dmt,consider(21)C1=∑s1s2∈EQG,s1∈VGs2∈VQG−VGdns1.
InC1, s1∈VG and dns1 occurs ds1 times. Thus(22)C1=∑s1s2∈EQG,s1∈VG,s2∈VQG−VGdn+1s1.
Let(23)C2=∑s1s2∈EQG,s1∈VG,s2∈VQG−VGdns2.
InC1, s2=uv∈EG and dns2 occurs two times. Therefore(24)C2=2∑s2=uv∈VQG−VGdu+dvn=2∑uv∈EG∑i=0nnidGn−iudGiv,∑2′=∑t∈VH∑s1s2∈EQG∑m=0nnm∑k=0mmkdn−m+ks1dmt+∑t∈VHdmt∑s1s2∈EQGdns2=∑m=0n∑k=0mnmmkM1α−m+kGM1mH+2∑m=0nnmM1mH∑uv∈G∑i=0nnidn−iudiv.
Now∀t1=t2∈VH and edge s1s2∈EQG where s1, s2∈VQG−VG, we have(25)∑2″=∑t1=t2∈VH∑s1,s2∈EQGdns1,t1+dns2,t2=∑t∈VH∑s1,s2∈EQGds1+ds1dtn+ds2+ds2dtn=∑t∈VH∑s1,s2∈EQG∑m=0nnmdn−ms1dms1dmt+∑m=0nnmdn−ms2dms2dmt=∑t∈VH∑s1,s2∈EQG∑m=0nnmdns1dmt+∑m=0nnmdns2dmt=∑t∈VH∑s1,s2∈EQG∑m=0nnmdmtdns1+dns2=∑t∈VH∑uv,vw∈EG∑m=0nnmdmtdu+dvn+dv+dwn=∑t∈VH∑uv,vw∈EG∑m=0nnmdmt∑i=0nnidn−iudiv+∑i=0nnidn−ivdiw=∑m=0nnm∑t∈VHdmt∑uv,vw∈EG∑i=0nnidn−iudiv+dn−ivdiw=∑m=0nnm∑t∈VHdmt∑uv,vw∈EG∑i=0nnidn−iudiv+dn−ivdiw=∑m=0nnm∑t∈VHdmt∑uv,vw∈EG∑i=0nnidn−iudiv+dn−ivdiw=∑m=0nnmM1mH∑uv,vw∈EG∑i=0nnidn−iudiv+dn−ivdiw.
For eacht1t2∈EH and s1s2∈EQG, where s1,s2∈VG, we have(26)∑3=∑s1s2∈EQG∑t1t2∈EHdns1,t1+dns2,t2∑3=∑s1s2∈EQGs1∈VG,s2∈VQG−VG∑t1t2∈EHdns1,t1+dns2,t2+∑s1s2∈EQGs1,s2∈VQG−VG∑t1t2∈EHdns1,t1+dns2,t2=∑3′+∑3″.∑3′=∑t1t2∈EH∑s1s2∈EQGdns1,t1+dns2,t2=∑t1t2∈EH∑s1s2∈EQGds1+dt1+ds1dt1n+ds2+ds2dt2n=∑t1t2∈EH∑s1s2∈EQG∑m=0nnm∑k=0mmkdn−ms1dm−kt1ds1dt1k+dn−ms2ds2dt2m=∑t1t2∈EH∑s1s2∈EQG∑m=0nnm∑k=0mmkdn−m+ks1dmt1+dns2dmt2=∑m=0nnm∑k=0mmkM1α−m+kGM1m+1H+∑t1t2∈EHdmt2∑s1s2∈EQGdns2=∑m=0nnm∑k=0mmkM1α−m+kGM1m+1H+2∑t1t2∈EHdmt2∑u,v∈VGdu+dvn=∑m=0nnm∑k=0mmkM1α−m+kGM1m+1H+2M1m+1H∑u,v∈VGdu+dvn=∑m=0nnm∑k=0mmkM1α−m+kGM1m+1H+2M1m+1H∑u,v∈VG∑i=0ndn−iudiv=∑m=0nnm∑k=0mmkM1α−m+kGM1m+1H+2M1m+1H∑i=0n∑u,v∈VGdn−iudiv.
For eacht1t2∈EH and s1s2∈EQG, where s1∈VG, s2∈VQG−VG, we have(27)∑3′′=∑t1t2∈EH∑s1s2∈EQGdns1,t1+dns2,t2=∑t1t2∈EH∑s1s2∈EQGds1+ds1dt1n+ds2+ds2dt2n=∑t1t2∈EH∑s1s2∈EQG∑m=0nnmdn−ms1ds1dt1m+∑m=0nnmdn−ms2ds2dt2m=∑m=0nnmM1m+1H∑u,v∈VGdu+dvn+∑v,w∈VGdv+dwn=∑m=0nnmM1m+1H∑i=0nni∑u,v∈VGdn−iudiv+∑v,w∈VGdn−iv+diw.
Hence(28)MαGQ⊠H=∑m=0n∑k=0mnmmkM1n+k−mGM1mHM1H+M1G+M1GM1H+∑m=0n∑i=0nnmni∑uv∈EGM1mHdGn−iudGiv2+2M1H+∑m=0nnm∑i=0nni∑uv,vw∈EGM1mHdGn−iudGiv+dGn−ivdGiw1+M1H.Theorem 4.
LetG and H be two graphs. For α∈N+ and n=α−1, FGZ index of T-sum graph is(29)MαGT⊠H=∑m=0n∑k=0mnmmk2n−m+kM1m+1HM1k−mGM1nG+3M1αG+∑m=0nnm∑uv∈VG∑i=0nnidGn−iudGiv3M1mH+3M1m+1H+∑m=0nnm∑uv∈VG∑i=0nnidGn−ivdGiw2M1m+1H+M1mH.Theorem 5.
LetG and H be two graphs. Then, the FGZI of Φ-sum graphs where α∈ℜ, we have(i)
MαGS⊠H=∑m=0∞∑k=0∞nmmkM1n−m+kGM1m+1H+M1α−m+kGM1mH+M1α−m+kGM1m+1H+2n+1eG∑m=0∞nmM1mH1+M1H,(ii)
MαGR⊠H=∑m=0∞∑k=0∞nmmk2n−m+kM1mHM1n−m+kGM1H+M1α−m+kG2+M1H+2∑m=0∞nm2n+1eGM1m+1H,(iii)
MαGQ⊠H=∑m=0∞∑k=0∞nmmkM1n+k−mGM1mHM1H+M1G+M1GM1H+∑m=0∞∑i=0∞nmni2+2M1H∑uv∈EGM1mHdGn−iudGiv+∑m=0∞nm∑i=0∞ni∑uv,vw∈EGM1mHdGn−iudGiv+dGn−ivdGiw1+M1H,(iv)
MαGT⊠H=∑m=0∞∑k=0∞nmmk2n−m+kM1m+1HM1k−mGM1nG+3M1αG+∑m=0∞nm∑uv∈VG∑i=0∞nidGn−iudGiv3M1mH+3M1m+1H+∑m=0∞nm∑vw∈VG∑i=0∞nidGn−ivdGiw2M1m+1H+M1mH.


## 4. Discussion and Conclusion

We conclude our work with applications of (Theorem1–Theorem 4) in the following examples.Example 1.
For two graphsPn and Pm of order n≥3 and m≥2, we have(30)MαPnS⊠Pm=∑m=0n∑k=0mCmnCkm2α+k+2n−2m−2+52α−m+kn−2+52m+1m−2+12+2n+1n−1∑m=0nCmn2m2m2−5m+2+4m−2,MαPnR⊠Pm=∑m=0nCmn∑k=0mCkm2n−m+k2m+1m+2+22α−m+kn−3+8+2n+3n−12mm−1+1,MαPnQ⊠Pm=∑m=0n∑k=0mCmnCkm2α+k+2n−2m−2+52α−m+kn−2+52m−2+12+∑m=0n∑i=0nCmnCin2mm−2+22i+2nn−2+2n−i4m−1+2i+2nn−2+2n−i+2m−1,MαPnT⊠Pm=∑m=0n∑k=0mCmnCkm2n−m+k2m+1m−2+22k−mn−2+22n+32α+∑m=0n∑i=0nCmnCin92mm−2+122i+2nn−2+2n−i+2m5m−10+62i+2nn−2+2n−i2.Example 2.
For two graphsCn and Cm of order n,m≥3, we have(31)MαCnS⊠Cm=∑m=0n∑k=0mCmnCkm2α+k+2nm+∑m=0nCmn2n+m+1mn−11+2mm,MαCnR⊠Cm=∑m=0n∑k=0mCmnCkm22n+2k−m+19mn+∑m=0nCmn2n+m+3mn−1,MαCnQ⊠Cm=∑m=0n∑k=0mCmnCkm2α+k+2nm+∑m=0n∑i=0nCmnCin2m+n+2nm1+2m,MαCnT⊠Cm=∑m=0n∑k=0mCmnCkm22n+k+1−mnm+∑m=0n∑i=0nCmnCin2n+m14nm.(i)
In this paper, we computed the exact values of FGZ index of the fourΦ-sum graphs (GS⊠H, GR⊠H, GQ⊠H and GT⊠H) in terms of their factor graphs, where these graphs are obtained under the operation of the strong product.(ii)
It is the extension of Deng et al. [13], Akhter and Imran [14], Liu et al. [15] and Sarala et al. [16] works.(iii)
The FGZ index of theΦ-sum graphs obtained from the paths and cycles are also computed as applications of the obtained results.(iv)
Table1 and Figure 3 present that FGZ index of PnT⊠Pm is dominant having ordering MαPnT⊠Pm≥MαPnR⊠Pm≥MαPnQ⊠Pm≥MαPnS⊠Pm.(v)
Table2 and Figure 4 also presents that FGZ index of CnT⊠Cm is dominant having ordering MαCnT⊠Cm≥MαCnR⊠Cm≥MαCnQ⊠Cm≥MαCnS⊠Cm.Table 1
Table for Example1.
[m, n, α]MαPnS⊠PmMαPnR⊠PmMαPnQ⊠PmMαPnT⊠Pm(3, 3, 3)1808641434428048(4, 4, 4)37260208104102800258412(5, 5, 5)434888550532218451826915616(6, 6, 6)5.42×10612785783628374140150810676(7, 7, 7)6.25×10727130459984063811623056832424(8, 8, 8)6.84×1075.39×10105604492565.88×1010(9, 9, 9)7.11×10101.02×10127.56×10101.088×1012(10, 10, 10)7.34×10101.85×10131.01×10121.94×1013Table 2
Table for Example2.
[m, n, α]MαCnS⊠CmMαCnR⊠CmMαCnQ⊠CmMαCnT⊠Cm(3, 3, 3)6552266401756837656(4, 4, 3)11648473603123266944(5, 5, 3)182007400048800104600Figure 3
Graphical representations for Table1.Figure 4
Graphical representations for Table2.



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