# Multiplicative Zagreb Indices of Molecular Graphs

Authors: Xiujun Zhang; H. M. Awais; M. Javaid; Muhammad Kamran Siddiqui
Journal: Journal of Chemistry
 (2019)

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## Abstract

Mathematical modeling with the help of numerical coding of graphs has been used in the different fields of science, especially in chemistry for the studies of the molecular structures. It also plays a vital role in the study of the quantitative structure activities relationship (QSAR) and quantitative structure properties relationship (QSPR) models. Todeshine et al. (2010) and Eliasi et al. (2012) defined two different versions of the 1st multiplicative Zagreb index as∏Γ=∏p∈VΓdΓp2 and ∏1Γ=∏pq∈EΓdΓp+dΓq, respectively. In the same paper of Todeshine, they also defined the 2nd multiplicative Zagreb index as ∏2Γ=∏pq∈EΓdΓp×dΓq. Recently, Liu et al. [IEEE Access; 7(2019); 105479–-105488] defined the generalized subdivision-related operations of graphs and obtained the generalized F-sum graphs using these operations. They also computed the first and second Zagreb indices of the newly defined generalized F-sum graphs. In this paper, we extend this study and compute the upper bonds of the first multiplicative Zagreb and second multiplicative Zagreb indices of the generalized F-sum graphs. At the end, some particular results as applications of the obtained results for alkane are also included.


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## Body

## 1. Introduction

The numerical demonstration of a molecular graph can be assumed as a single number, commonly known as topological index (TI). There are many interesting and significant results about TIs to study the different properties of chemical compounds such as chromatographic retention times, heat of formation and evaporation, flash point, viscosity, freezing, boiling and melting point, octanol-water partition coefficient, surface tension, stability, temperature, density, weight, polarizability, connectivity, and solubility. Many medicines, crystalline and nanomaterials, that are used in numerous pharmaceutical industries are examined with the assistance of different TIs, see [1–7]. TIs also study QSPR and QSAR models that join molecular graphs to their molecular characteristics by means of statistical tools. For additional information, see [8–17].In 1947, to investigate the paraffin’s boiling point, Wiener utilized the distance-based TI [18]. Gutman and Trinajstić [19] determined a pair of degree-based first and second Zagreb indices. After this, different impressive TIs are introduced in molecular graph theory [20,21], but the degree-based TIs are famous than others, see [22].In graph theory, the various operations on different graphs show an important role in the creation of advanced families of graphs, see [23,24]. Yan et al. [13] gave the idea of four operations S1,R1,Q1, and T1 of graphs and computed the Wiener index of the resultant graphs obtained by using these operations. Eliasi and Taeri [25] introduced the F1-sum graphs Γ1+F1Γ2 by using the cartesian product of graphs F1Γ1 and Γ2, where Γ1 and Γ2 are the two simple graphs and F1Γ1 is obtained by using F1∈S1,R1,Q1,&T1. They also determined the Wiener index of these F1-sum graphs. Additionally, Deng et al. [26], Imran and Akhtar [27], Shirdel et al. [28], and Liu et al. [29] determined the 1st and 2nd Zagreb indices, F-index, Hyper-Zagreb index, and the 1st general Zagreb index of F1-sum graphs.Recently, Liu et al. [30] defined the generalized form of the aforesaid four operations Sk,Rk,Qk,andTk, fork≥1integer. They also constructed the generalized F-sum graphs (Γ1+FkΓ2 for Fk∈Sk,Rk,Qk,&Tk) and obtained the 1st and 2nd Zagreb indices. In this paper, we calculate the upper bounds of the 1st and 2nd multiplicative Zagreb indices of the Fk-sum graphs. The remaining article is organized as follows: Section 2 contains few definitions and terminologies, Section 3 covers the main results, and Section 4 includes applications and closing comments.


## 2. Preliminaries

A molecular graphΓ=VΓ,EΓ has the node (vertex) set VΓ=p1,p2,p3,…,pn and the edge set EΓ⊆VΓ×VΓ. The vertices in the molecular graphs are denoted as atoms, and bonds are denoted as edges. The order and size of a chemical structure is denoted as VΓ=μ and EΓ=ω. The degree dΓp is the total edges that are incident on a node p. Throughout the paper, we study the finite, indirected, simple (without loops, isolated vertices, and multiple edges), and connected graphs. The well-known TIs are discussed as follows.Definition 1.
LetΓ be a graph, then the first and second Zagreb indices are defined as follows:(1)M1Γ=∑pq∈EΓdΓp+dΓq,M2Γ=∑pq∈EΓdΓp×dΓq.
Gutman and Trinajstić [19] defined these indices which are utilized to determine the structural base different characteristics of graphs like molecular complexity, energy, ZE-isomerism, chirality, connectivity, and heterosystems, as well as branching , see [15,31–36].Definition 2.
LetΓ be a graph, then the first and second multiplicative Zagreb indices are(2)∏1Γ=∏pq∈EΓdΓp+dΓq,∏2Γ=∏pq∈EΓdΓp×dΓq.
During the past two decades, various noteworthy applications regarding multiplicative Zagreb indices have been explained in detail, see [37–46] and the references cited therein. The resultant graphs under the four new generalized subdivision operations defined in [30] are given as follows:(i)
SkΓ is the generalized subdivided graph Γ.(ii)
RkΓ is the generalized semitotal (point) graph.(iii)
QkΓ is the generalized semitotal (line) graph.(iv)
TkΓ is the generalized total graph. For more details, see Figure 1.Figure 1
(a)Γ, (b) S4Γ, (c) R4Γ. (d) Q4Γ, and (e) T4Γ.
(a)
(b)
(c)
(d)
(e)Definition 3.
LetΓ1 and Γ2 be two graphs, thenFk∈Sk,Rk,Qk,Tk and FkΓ1 be a new graph gained after using Fk on Γ1 having an edge set EFkΓ1 and vertex set VFkΓ1. So, the generalized F-sum graph (Γ1+FkΓ2) is a graph having the vertex set VΓ1+FkΓ2=VFkΓ1×VΓ2=VΓ1∪EΓ1×VΓ2 in such a way that two vertices p1,q1 and p2,q2 of VΓ1+FkΓ2 are adjacent if and only if [p1=p2∈VΓ1 and q1,q2∈EΓ2] or [q1=q2∈Γ2 and p1,p2∈EFkΓ1].
Thus, the generalizedF-sum graph Γ1+FkΓ2 contains VΓ2 copies of new graphs FkΓ1 that are labeled with the vertex set of Γ2. For further details, see Figures 2 and 3.Figure 2
(a)Γ1≅C3, (b) Γ2≅C3, (c) Γ1+S2Γ2, and (d) Γ1+R2Γ2.
(a)
(b)
(c)
(d)Figure 3
(a)H1+Q2H2 and (b) Γ1+T2Γ2.
(a)
(b)


## 3. Main Results

Now, we calculate the key results of the multiplicative Zagreb indices for the different classes on graphs.Theorem 1.
LetΓ1, Γ2 be two graphs with VΓ1=μ1, VΓ2=μ2, EΓ1=ω1, and EΓ2=ω2. For μ1,μ2≥4and and k≥1, we have(3)∏1Γ1+ΓSk2≤1μ1ω24ω1ω2+μ1M1Γ2μ1ω2×12ω1μ24ω1ω2+μ2M1S1Γ12ω1μ2×1ω1μ2k−14k−1μ2ω1ω1μ2k−1,∏2Γ1+ΓSk2≤1μ1ω2ω2M1Γ1+2ω1M1Γ2+μ1M2Γ2μ1ω2×12μ2ω1μ2M2S1Γ1+8ω1ω22μ2ω1×1k−1μ2ω14k−1μ2ω1k−1μ2ω1.Proof.
(a)
Considerdp,q=dΓ1+SkΓ2p,q as a degree of a node p,q in Γ1+ΓSk2.
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Now,(5)∏1=∏p∈VΓ1∏q1q2∈EΓ2dp,q1+dp,q2≤1μ1ω2∑p∈VΓ1∑q1q2∈EΓ2dp,q1+dp,q2μ1ω2=1μ1ω2∑p∈VΓ1∑q1q2∈EΓ22dΓ1p+dΓ2q1+dΓ2q2μ1ω2=1μ1ω24ω1ω2+μ1M1Γ2μ1ω2,∏2=∏q∈VΓ2∏p1p2∈ESkΓ1dp1,q+dp2,q≤12ω1μ2∑q∈VΓ2∑p1p2∈ESkΓ1p1∈VΓ1,p2∈VSkΓ1−μΓ1dp1,q+dp2,q2ω1μ2=12ω1μ2∑q∈VΓ2∑p1p2∈ESkΓ1p1∈VΓ1,p2∈VSkΓ1−μΓ1dSkΓ1p1+dΓ2q+dSkΓ1p22ω1μ2=12ω1μ24ω1ω2+μ2M1S1Γ12ω1μ2,∏3=∏q∈VΓ2∏p1p2∈ESkΓ1p1,p2∈VSkΓ1−Γ1dp1,q+dp2,q≤1ω1μ2k−1k−1∑q∈VΓ2∑p1p2∈ESkΓ1p1,p2∈VSkΓ1−μΓ1dp1,q+dp2,qω1μ2k−1=1ω1μ2k−1k−1∑q∈VΓ2∑p1p2∈ESkΓ1p1,p2∈VSkΓ1−μΓ12+2ω1μ2k−1=1ω1μ2k−1k−1ωΓ1∑q∈VΓ24ω1μ2k−1=1ω1μ2k−14k−1μ2ω1ω1μ2k−1.
Hence,(6)∏1Γ1+ΓSk2≤1μ1ω24ω1ω2+μ1M1Γ2μ1ω2×12ω1μ24ω1ω2+μ2M1S1Γ12ω1μ2×1ω1μ2k−14k−1μ2ω1ω1μ2k−1,b∏2Γ1+ΓSk2=∏p1,q1p2,q2∈EΓ1+SkΓ2dp1,q1dp2,q2=∏p∈VΓ1∏q1q2∈EΓ2dp,q1dp,q2×∏q∈VΓ2∏p1p2∈ESkΓ1p1,p2∈VSkΓ1−μΓ1dp1,qdp2,q4×∏q∈VΓ2∏p1p2∈ESkΓ1p1∈VΓ1,p2∈VSkΓ1−μΓ1dp1,qdp2,q=∏1×∏2×∏3.
Now,(7)∏1=∏p∈VΓ1∏q1q2∈EΓ2dp,q1dp,q2≤1μ1ω2∑p∈VΓ1∑q1q2∈EΓ2dp,q1dp,q2μ1ω2=1μ1ω2∑p∈VΓ1∑q1q2∈EΓ2dΓ1p+dΓ2q1dΓ1p+dΓ2q2μ1ω2=1μ1ω2ω2M1Γ1+2ω1M1Γ2+μ1M2Γ2μ1ω2,∏2=∏q∈VΓ2∏p1p2∈ESkΓ1p1∈VΓ1,p2∈VSkΓ1−μΓ1dp1,qdp2,q≤12μ2ω1∑q∈VΓ2∑p1p2∈ESkΓp1∈VΓ1,p2∈VSkΓ1−μΓ1dp1,qdp2,q2μ2ω1=12μ2ω1∑q∈VΓ2∑p1p2∈ESkΓ1p1∈VΓ1,p2∈VSkΓ1−μΓ1dSkΓ1p1+dΓ2qdSkΓ1p22μ2ω1,∏3=∑q∈VΓ2∑p1p2∈ESkΓ1p1∈VΓ1,p2∈VSkΓ1−μΓ1dp1,qdp2,q≤1k−1μ2ω1∏q∈VΓ2∏p1p2∈ESkΓ1p1,p2∈VSkΓ1−μΓ1dp1,qdp2,qk−1μ2ω1=1k−1μ2ω1∑q∈VΓ2∑p1p2∈ESkΓ1p1,p2∈VSkΓ1−μΓ14k−1μ2ω1=1k−1μ2ω14k−1μ2ω1k−1μ2ω1.
Consequently,(8)∏2Γ1+ΓSk2≤1μ1ω2ω2M1Γ1+2ω1M1Γ2+μ1M2Γ2μ1ω2×12μ2ω1μ2M2S1Γ1+8ω1ω22μ2ω1×1k−1μ2ω14k−1μ2ω1k−1μ2ω1.Theorem 2.
LetΓ1, Γ2 be two graphs with VΓ1=μ1, VΓ2=μ2, EΓ1=ω1, and EΓ2=ω2. For μ1,μ2≥4 and k≥1, we have(9)∏1Γ1+ΓRk2≤1μ1ω2μ1M1Γ2+8ω1ω2μ1ω2×1ω1μ24ω1ω2+2μ2M1Γ1ω1μ2×12ω1μ24ω1ω2+μ2M1R1Γ1−2M1Γ12ω1μ2×1ω1μ2k−14k−1μ2ω1ω1μ2k−1,∏2Γ1+ΓRk2≤1ω2μ14ω2M1Γ1+4ω1M1Γ2+μ1M2Γ2ω2μ1×1ω1μ24μ2M2Γ1+4ω2M1Γ1+ω1M1Γ2ω1μ2×12ω1μ28ω1ω2+μ2M2R1Γ1−4M2Γ12ω1μ2×1k−1ω1μ24μ2ω1k−1k−1ω1μ2.Proof.
(a)
Considerdp,q=dΓ1+ΓRk2p,q as a degree of a node p,q in Γ1+RkΓ2:
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Now,(11)∏1=∏p∈VΓ1∏q1q2∈EΓ2dp,q1+dp,q2≤1μ1ω2∑p∈VΓ1∑q1q2∈EΓ2dp,q1+dp,q2μ1ω2=1μ1ω2∑p∈VΓ1∑q1q2∈EΓ22dRkΓ1p+dΓ2q1+dΓ2q2μ1ω2=1μ1ω2∑p∈VΓ1∑q1q2∈EΓ24dΓ1p+dΓ2q1+dΓ2q2μ1ω2=1μ1ω2μ1M1Γ2+8ω1ω2μ1ω2,∏2=∏q∈VΓ2∏p1p2∈ERkΓ1dp1,q+dp2,q=∏q∈VΓ2∏p1p2∈ERkΓ1p1,p2∈VΓ1dp1,q+dp2,q×∏q∈VΓ2∏p1p2∈ERkΓ1p1∈VΓ1dp1,q+dp2,q×∏q∈VΓ2∏p1p2∈ERkΓ1p1,p2∈VRkΓ1−μΓ1dp1,q+dp2,q=∏′2×∏′′2×∏′′′2.
Now, forp1p2∈VΓ1, then p1p2∈ERkΓ1 if and only if p1p2∈EΓ1; for p1∈VΓ1, we get dRkΓ1p1=2dΓ1p1 and for p2∈VRkΓ1−μΓ1, we have dRkΓ1p2=2. Now,(12)∏′2=∏q∈VΓ2∏p1p2∈ERkΓ1p1,p2∈VΓ1dp1,q+dp2,q≤1ω1μ2∑q∈VΓ2∑p1p2∈ERkΓ1p1p2∈VΓ1dp1,q+dp2,qω1μ2=1ω1μ2∑q∈VΓ2∑p1p2∈EΓ12dΓ2q+∑q∈VΓ2∑p1p2∈EΓ1dRkΓ1p1+dRkΓ1p2ω1μ2=1ω1μ24ω1ω2+2μ2M1Γ1ω1μ2,∏″2=∏q∈VΓ2∏p1p2∈ERkΓ1p1∈VΓ1dp1,q+dp2,q≤12ω1μ2∑q∈VΓ2∑p1p2∈ERkΓ1p1∈VΓ1dp1,q+dp2,q2ω1μ2=12ω1μ2∑q∈VΓ2∑p1p2∈ERkΓ1p1∈VΓ1dRkΓ1p1+dΓ2q+dRkΓ1p22ω1μ2=12ω1μ24ω1ω2+μ2M1R1Γ1−2M1Γ12ω1μ2,∏″′2=∏q∈VΓ2∏p1p2∈ERkΓ1p1,p2∈VRkΓ1−μΓ1dp1,q+dp2,q≤1ω1μ2k−1∑q∈VΓ2∑p1p2∈ERkΓ1p1,p2∈VRkΓ1−μΓ1dp1,q+dp2,qω1μ2k−1=1ω1μ2k−14k−1μ2ω1ω1μ2k−1.
Hence,(13)∏1Γ1+ΓRk2≤1μ1ω2μ1M1Γ2+8ω1ω2μ1ω2×1ω1μ24ω1ω2+2μ2M1Γ1ω1μ2×12ω1μ24ω1ω2+μ2M1R1Γ1−2M1Γ12ω1μ2×1ω1μ2k−14k−1μ2ω1ω1μ2k−1,b∏2Γ1+RkΓ2=∏p1,q1p2,q2.∈EΓ1+RkΓ2dp1,q1dp2,q2=∏p∈VΓ1∏q1q2∈EΓ2dp,q1dp,q2×∏q∈VΓ2∏p1p2∈ERkΓ1dp1,qdp2,q=∏1×∏2.
Now,(14)∏1=∏p∈VΓ1∏q1q2∈EΓ2dp,q1dp,q2≤1ω2μ1∑p∈VΓ1∑q1q2∈EΓ2dp,q1dp,q2ω2μ1=1ω2μ1∑p∈VΓ1∑q1q2∈EΓ2dRkΓ1p+dΓ2q1dRkΓ1p+dΓ2q2ω2μ1=1ω2μ14ω2M1Γ1+4ω1M1Γ2+μ1M2Γ2ω2μ1,∏2=∏q∈VΓ2∏p1p2∈ERkΓ1dp1,qdp2,q=∏q∈VΓ2∏p1p2∈ERkΓ1p1,p2∈VΓ1dp1,qdp2,q×∏q∈VΓ2∏p1p2∈ERkΓ1p1∈VΓ1p2∈VRkΓ1−μΓ2dp1,qdp2,q×∏q∈VΓ2∏p1p2∈ERkΓ1p1,p2∈VRkΓ1−μΓ1dp1,qdp2,q=∏′2×∏″2×∏″′2.
Now, we consider(15)∏′2=∏q∈VΓ2∏p1p2∈ERkΓ1p1p2∈VΓ1dp1,qdp2,q=∏q∈VΓ2∏p1p2∈EΓ1dp1,qdp2,q≤1ω1μ2∑q∈VΓ2∑p1p2∈EΓ1dp1,qdp2,qω1μ2=1ω1μ2∑q∈VΓ2∑p1p2∈EΓ1dRkΓ1p1+dΓ2qdRkΓ1p2+dΓ2qω1μ2=1ω1μ24μ2M2Γ1+4ω2M1Γ1+ω1M1Γ2ω1μ2,∏″2=∏q∈VΓ2∏p1p2∈ERkΓ1p1∈VΓ1dp1,qdp2,q≤12ω1μ2∑q∈VΓ2∑p1p2∈ERkΓ1p1∈VΓ1dp1,qdp2,q2ω1μ2=12ω1μ2∑q∈VΓ2∑p1p2∈ERkΓ1p1∈VΓ1dRkΓ1p1+dΓ2q×22ω1μ2=12ω1μ28ω1ω2+μ2M2R1Γ1−4M2Γ12ω1μ2,∏″′2=∏q∈VΓ2∏p1p2∈ERkΓ1p1p2∈VRkΓ1−μΓ1dp1,qdp2,q≤1k−1ω1μ2k−1∑q∈VΓ2∑p1p2∈ERkΓ1p1p2∈VRkΓ1−μΓ1dp1,qdp2,qk−1ω1μ2=1k−1ω1μ24μ2ω1k−1k−1ω1μ2.
Therefore, we have(16)∏2Γ1+ΓRk2≤1ω2μ14ω2M1Γ1+4ω1M1Γ2+μ1M2Γ2ω2μ1×1ω1μ24μ2M2Γ1+4ω2M1Γ1+ω1M1Γ2ω1μ2×12ω1μ28ω1ω2+μ2M2R1Γ1−4M2Γ12ω1μ2×1k−1ω1μ24μ2ω1k−1k−1ω1μ2.Theorem 3.
LetΓ1, Γ2 be two graphs with VΓ1=μ1, VΓ2=μ2, EΓ1=ω1, and EΓ2=ω2. For μ1,μ2≥4 and and k≥1, we have(17)∏1Γ1+ΓQk2≤1μ1ω24ω1ω2+μ1M1Γ2μ1ω2×12ω1μ24ω1ω2+3μ2M1Γ12ω1μ2×1ω1μ2k−12k−1μ2M1Γ1ω1μ2k−1×1kμ22ω1−μ1kμ2M3Γ1+2M2Γ1−2M1Γ1kμ22ω1−μ1,∏2Γ1+ΓQk2≤1μ1ω2ω2M1Γ1+2ω1M1Γ2+μ1M2Γ2μ1ω2×12μ2ω14ω2M1Γ1+μ2M3Γ1+2M2Γ12μ2ω1×1k−1μ2ω1k−1μ22M2Γ1+M3Γ1k−1μ2ω1×1kμ22ω1−μ1kμ212M4Γ1−12M3Γ1+∑uv∈VΓ1rdΓ1udΓ1vkμ22ω1−μ1+1kμ22ω1−μ1∑v∈VΓ1dΓ12v∑u∈VΓ1uv∈EΓ1dΓ1u−2M2Γ1kμ22ω1−μ1.Proof.
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2
.
Now,(19)∏1=∏p∈VΓ1∏q1q2∈EΓ2dp,q1+dp,q2≤1μ1ω2∑p∈VΓ1∑q1q2∈EΓ22dΓ1p+dΓ2q1+dΓ2q2μ1ω2=1μ1ω24ω1ω2+μ1M1Γ2μ1ω2,∏2=∏q∈VΓ2∏p1p2∈EQkΓ1dp1,q+dp2,q=q∈VΓ2∏q∈VΓ2∏p1p2∈EQkΓ1p1∈VΓ1p1p2∈VQkΓ1−μΓ1dp1,q+dp2,q×∏q∈VΓ2∏p1p2∈EQkΓ1p1p2∈VQkΓ1−μΓ1dp1,q+dp2,q=∏′2×∏″2.
Consider(20)∏′2=∏q∈VΓ2∏p1p2∈EQkΓ1p1∈VΓ1p1p2∈VQkΓ1−μΓ1dp1,q+dp2,q≤12ω1μ2∑q∈VΓ2∑p1p2∈EQkΓ1p1∈VΓ1p1p2∈VQkΓ1−μΓ1dQkΓ1p1+dΓ2q+dQkΓ1p22ω1μ2=12ω1μ2∑q∈VΓ2∑p1∈VΓ1dΓ1p1+dΓ2q2ω1μ2+12ω1μ2∑q∈VΓ2∏p1p2∈EQkΓ1p1∈VΓ1p1p2∈VQkΓ1−μΓ1dQkΓ1p22ω1μ2.
Then,(21)∏′2=12ω1μ24ω1ω2+3μ2M1Γ12ω1μ2.
Now we take(22)∏″2=∏q∈VΓ2∏p1p2∈EQkΓ1p1p2∈VQkΓ1−μΓ1dp1,q+dp2,q.
Let∏″2=∏3×∏4, then(23)∏3=∏q∈VΓ2∏p1p2∈EQkΓ1p1p2∈VQkΓ1−μΓ1dQkΓ1p1+dQkΓ1p2≤1ω1μ2k−12k−1∑q∈VΓ2∑uv∈EΓ1dΓ1u+dΓ1vω1μ2k−1=1ω1μ2k−12k−1μ2M1Γ1ω1μ2k−1,∏4=∏q∈VΓ2∏p1p2∈EQkΓ1p1p2∈VQkΓ1−μΓ1dQkΓ1p1+dQkΓ1p2≤1kμ22ω1−μ1∑q∈VΓ2∑p1p2∈EQkΓ1p1p2∈VQkΓ1−μΓ1dQkΓ1p1+dQkΓ1p2kμ22ω1−μ1=1kμ22ω1−μ1kμ2M3Γ1+2M2Γ1−2M1Γ1kμ22ω1−μ1.
Therefore, we have(24)∏1Γ1+ΓQk2≤1μ1ω24ω1ω2+μ1M1Γ2μ1ω2×12ω1μ24ω1ω2+3μ2M1Γ12ω1μ2×1ω1μ2k−12k−1μ2M1Γ1ω1μ2k−1×1kμ22ω1−μ1kμ2M3Γ1+2M2Γ1−2M1Γ1kμ22ω1−μ1.
Next,(25)∏2Γ1+ΓQk2=∏p1,q1p2,q2∈EΓ1+QkΓ2dp1,q1+dp2,q2=∏p∈VΓ1∏q1q2∈EΓ2dp,q1dp,q2×∏q∈VΓ2∏p1p2∈EQkΓ1dp1,qdp2,q=∏1×∏2.
Consider(26)∏1=∏p∈VΓ1∏q1q2∈EΓ2dp,q1dp,q2≤1μ1ω2∑p∈VΓ1∑q1q2∈EΓ2dp,q1dp,q2μ1ω2=1μ1ω2ω2M1Γ1+2ω1M1Γ2+μ1M2Γ2μ1ω2,∏2=∏q∈VΓ2∏p1p2∈EQkΓ1dp1,qdp2,q=∏q∈VΓ2∏p1p2∈EQkΓ1p1∈VΓ1dp1,qdp2,q×∏q∈VΓ2∏p1p2∈EQkΓ1p1p2∈VQkΓ1−μΓ1dp1,qdp2,q=∏′2×∏″2.
Then,(27)∏′2=∏q∈VΓ2∏p1p2∈EQkΓ1p1∈VΓ1dp1,qdp2,q≤12ω1μ2∑q∈VΓ2∑p1p2∈EQkΓ1p1∈VΓ1dp1,qdp2,q2ω1μ2=12ω1μ2∑q∈VΓ2∑p1p2∈EQkΓ1p1∈VΓ1dΓ1p1+dΓ2qdQkΓ1p22ω1μ2=12ω1μ24ω2M1Γ1+μ2M3Γ1+2M2Γ12ω1μ2,∏″2=∏q∈VΓ2∏p1p2∈EQkΓ1p1p2∈VQkΓ1−μΓ1dp1,qdp2,q.
Let∏″2=∏3×∏4, then(28)∏3=∏q∈VΓ2∏p1p2∈EQkΓ1p1p2∈VQkΓ1−μΓ1dQkΓ1p1dQkΓ1p2≤1k−1μ2ω1∑q∈VΓ2∑p1p2∈EQkΓ1p1p2∈VQkΓ1−μΓ1dQkΓ1p1dQkΓ1p2k−1μ2ω1=1k−1μ2ω1k−1μ22M2Γ1+M3Γ1k−1μ2ω1,∏4=∏q∈VΓ2∏p1p2∈EQkΓ1p1p2∈VQkΓ1−μΓ1dQkΓ1p1dQkΓ1p2≤1kμ22ω1−μ1∑q∈VΓ2∑p1p2∈EQkΓ1p1p2∈VQkΓ1−μΓ1dQkΓ1p1dQkΓ1p2kμ22ω1−μ1=1kμ22ω1−μ1k∑q∈VΓ2∑uv∈EΓ1vw∈EΓ2dΓ1u+dΓ1vdΓ1v+dΓ1wkμ22ω1−μ1=1kμ22ω1−μ1kμ212M4Γ1−12M3Γ1+∑uv∈VΓ1rdΓ1udΓ1vkμ22ω1−μ1+1kμ22ω1−μ1∑v∈VΓ1dΓ12v∑u∈VΓ1uv∈EΓ1dΓ1u−2M2Γ1kμ22ω1−μ1,where r is the total neighbors that we consider mutual nodes of u and v in Γ1.
Therefore,(29)∏2Γ1+Γ2Qk≤1μ1ω2ω2M1Γ1+2ω1M1Γ2+μ1M2Γ2μ1ω2×12μ2ω14ω2M1Γ1+μ2M3Γ1+2M2Γ12μ2ω1×1k−1μ2ω1k−1μ22M2Γ1+M3Γ1k−1μ2ω1×1kμ22ω1−μ1kμ212M4Γ1−12M3Γ1+∑uv∈VΓ1rdΓ1udΓ1vkμ22ω1−μ1+1kμ22ω1−μ1∑v∈VΓ1dΓ12v∑u∈VΓ1uv∈EΓ1dΓ1u−2M2Γ1kμ22ω1−μ1.Theorem 4.
LetΓ1, Γ2 be two graphs with VΓ1=μ1, VΓ2=μ2, EΓ1=ω1, and EΓ2=ω2. For μ1,μ2≥4 and and k≥1, we have(30)∏1Γ1+ΓTk2≤1μ1ω2μ1M1Γ2+8ω1ω2μ1ω2×1ω1μ24ω1ω2+2μ2M1Γ1ω1μ2×12ω1μ24ω1ω2+4μ2M1Γ12ω1μ2×1ω1μ2k−12k−1μ2M1Γ1ω1μ2k−1×1kμ22ω1−μ1kμ2M3Γ1+2M2Γ1−2M1Γ1kμ22ω1−μ1,∏2Γ1+TkΓ2≤1ω2μ14ω2M1Γ1+4ω1M1Γ2+μ1M2Γ2ω2μ1×1ω1μ24μ2M2Γ1+4ω2M1Γ1+ω1M1Γ2ω1μ2×12μ2ω14ω2M1Γ1+μ22M3Γ1+4M2Γ12μ2ω1×1k−1μ2ω1k−1μ22M2Γ1+M3Γ1k−1μ2ω1×1kμ22ω1−μ1kμ212M4Γ1−12M3Γ1+∑uv∈VΓ1rdΓ1udΓ1vkμ22ω1−μ1+1kμ22ω1−μ1∑v∈VΓ1dΓ12v∑u∈VΓ1uv∈EΓ1dΓ1u−2M2Γ1kμ22ω1−μ1.


## 4. Conclusion

Fora,b≥4 and k=4, we consider that Γ1=Pa and Γ2=Pb are specific examples of alkane known as paths having orders a and b, respectively. Then, the following results are the direct outcomes of the above four theorems.(31)∏1Pa+S4Pb=1ab−a8ab−10a−4a+4ab−a×12ab−2b12ab−4a−14b+42ab−2b×13ab−3b12ab−12b3ab−3b,∏2Pa+S4Pb=1ab−a16ab−24a−14b+18ab−a×12ab−2b16ab−8a−20b+82ab−2b×13ab−3b12ab−12b3ab−3b,∏1Pa+R4Pb=1ab−a12ab−14a−8b+8ab−a×1ab−b12ab−4a−16b+4ab−b×12ab−2b16ab−4a−20b+42ab−2b×13ab−3b12ab−12b3ab−3b,∏2Pa+R4Pb=1ab−a36ab−48a−40b+48ab−a×1ab−b36ab−22a−60b+30ab−b×12ab−2b24ab−8a−32b+82ab−2b×13ab−3b12ab−12b3ab−3b,∏1Pa+Q4Pb=1ab−a8ab−10a−4b+4ab−a×12ab−2b16ab−4a−22b+42ab−2b×13ab−3b24ab−36b3ab−3b×14ab−8b32ab−72b4ab−8b,∏2Pa+Q4Pb=1ab−a16ab−24a−14b+18ab−a×12ab−2b32ab−16a−54b+242ab−2b×13ab−3b48ab−90b3ab−3b×14ab−8b64ab−160b4ab−8b,∏1Pa+T4Pb=1ab−a12ab−14a−8b+8ab−a×1ab−b12ab−4a−16b+4ab−b×12ab−2b20ab−4a−28b+42ab−2b×14ab−8b24ab−36b4ab−8b×13ab−3b32ab−72b3ab−3b,∏2Pa+T4Pb=1ab−a36ab−48a−40b+48ab−a×1ab−b36ab−22a−60b+30ab−b×12ab−2b48ab−16a−84b+242ab−2b×14ab−8b48ab−90b4ab−8b×13ab−3b64ab−160b3ab−3b,In this paper, for the integerk≥1, we computed upper bonds of the 1st and 2nd multiplicative Zagreb indices of the generalized F-sum graphs [Γ1+FkΓ2 for Fk∈Sk,Rk,Qk,&Tk] that was defined by Liu et al. [30]. However, the problem is still open to compute the other topological indices of the generalized F-sum graphs.



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*Source: 5294198-2019-12-06.xml*