# First General Zagreb Index of GeneralizedF-sum Graphs

Authors: H. M. Awais; Muhammad Javaid; Akbar Ali
Journal: Discrete Dynamics in Nature and Society
 (2020)

---

## Abstract

The first general Zagreb (FGZ) index (also known as the general zeroth-order Randić index) of a graphG can be defined as MγG=∑uv∈EGdGγ−1u+dGγ−1v, where γ is a real number. As MγG is equal to the order and size of G when γ=0 and γ=1, respectively, γ is usually assumed to be different from 0 to 1. In this paper, for every integer γ≥2, the FGZ index Mγ is computed for the generalized F-sums graphs which are obtained by applying the different operations of subdivision and Cartesian product. The obtained results can be considered as the generalizations of the results appeared in (IEEE Access; 7 (2019) 47494–47502) and (IEEE Access 7 (2019) 105479–105488).


---

## Body

## 1. Introduction

Graph theory concepts are being utilized to model and study the several problems in different fields of science, including chemistry and computer science. A topological index (TI) of a (molecular) graph is a numeric quantity that remained unchanged under graph isomorphism [1,2]. Many topological indices have found applications in chemistry, especially in the quantitative structure-activity/property relationships studies; for detail, see [3–13].Wiener index is the first TI introduced by Harry Wiener in 1947, when he was working on the boiling point of paraffin [14]. In 1972, Trinajstić and Gutman [15] obtained a formula concerning the total energy of π electrons of molecules where the sum of square of valences of the vertices of a molecular structure was appeared. This sum is nowadays known as the first Zagreb index. In this paper, we are concerned with a generalized version of the first Zagreb index, known as the general first Zagreb index as well as the general zeroth-order Randić index.There are several operations in graph theory such as product, complement, addition, switching, subdivision, and deletion. In many cases, graph operations may be helpful in finding graph quantities of more complicated graphs by considering the less complicated ones. In chemical graph theory, by using different graph operations, one can develop large molecular structures from the simple and basic structures. Recently, many classes of molecular structures are studied with the assistance of graph operations.In 2007, Yan et al. [6] listed the five subdivision operations with the help of their vertices and edges. They also discussed the different features of Wiener index of graphs under these operations. After that, Eliasi and Taeri [16] introduced the F1-sum graphs Γ1+F1Γ2 with the assistance of Cartesian product on graphs F1Γ1 and Γ2, where F1Γ1 is obtained by applying the subdivision operations S1,R1,Q1, and T1. They also defined the Wiener indices of these resulting graphs Γ1+S1Γ2, Γ1+R1Γ2, Γ1+Q1Γ2, and Γ1+T1Γ2. Later on, Deng et al. [17] calculated the 1st and 2nd Zagreb topological indices, and Imran and Akhtar [18] calculated the forgotten topological index of the F1-sums graph. In 2019, Liu et al. [19] computed the first general Zagreb index of F1-sums graphs.Recently, Liu et al. [20] introduced the generalized version of the aforesaid subdivided operations of graphs denoted by Sk,Rk,Qk,andTk, where k≥1 is counting number. They also defined the generalized F-sums graphs using these generalized operations and calculated their 1st and 2nd Zagreb indices. In the present work, we compute the 1st general Zagreb index of the generalized F-sums graphs Γ1+FkΓ2 for Fk∈Sk,Rk,Qk,Tk. The remaining work is arranged as follows: Section 2 contains some basic definitions, Section 3 contains the key outcomes, and Section 4 contains the some particular applications. Conclusions of the obtained results are presented in Section 5.


## 2. Preliminaries

LetΓ=VΓ,EΓ be a simple graph having VΓ the order andEΓ the size of a graph, where VΓ is considered as node set and EΓ⊆VΓ×VΓ is a bond set. Every vertex is considered as an atom in a graph, and bonding within the two atoms is known as edge. The valency or degree of any node is the number of total edges which are incident to the node. Now, few useful TI’s are explained given below:Definition 1.
IfΓ be a connected graph, then the 1st and 2nd Zagreb topological indices as(1)M1Γ=∑uv∈VΓdΓu+dΓv,M2Γ=∑uv∈VΓdΓudΓv.
These two descriptors of the graph were introduced by Trinajsti and Gutman [15]. Such type of TI’s have been utilized to discuss the QSAR/QSPR of the different chemical structures such as chirality, complexity, hetero-system, ZE-isomers, π electron energy, and branching [9, 10].Definition 2.
IfR is the real number, γ∈R−0,1, and Γ be a connected graph, so the 1st general Zagreb topological index is given as(2)MγΓ=∑uv∈EΓdΓγ−1u+dΓγ−1v.Definition 3.
IfR is the real number, γ∈R, and Γ be a connected graph, so the general Randic´ is given as(3)RγΓ=∑uv∈EΓdGudvγ,where R−1/2 is considered as the classical Randic´ connectivity topological index.
The generalizedF-sums graph is defined in [20] as follows:(i)
SkG graph is obtained by inserting k vertices in each edge of G.(ii)
RkG is obtained from SkG by joining the old vertices which are adjacent G.(iii)
QkG is obtained from SkG by joining the new vertices lying on edge to the corresponding new vertices of other edge, if these edges have some common vertex in G.(iv)
TkG is union of RkG and QkG graphs. For further details, see Figure 1.Figure 1
(a)Γ, (b). S2Γ, (c) R2Γ, (d) Q2Γ, and (e) T2Γ.
(a)(b)(c)(d)(e)Definition 4.
IfΓ1&Γ2 be two connected molecular structures, Fk∈Sk,Rk,Qk,Tk and FkΓ1 be a structure obtained after using Fk on Γ1 with bonds (edges) EFkΓ1 and nodes (vertices) VFkΓ1. So, the generalized F-sums graph (Γ1+FkΓ2) is a structure with nodes:(4)VΓ1+FkΓ2=VFkΓ1×VΓ2,=VΓ1∪EΓ1×VΓ2,in such a way two nodes a1,b1&a2,b2 of VΓ1+FkΓ2 are adjacent if [a1=a2∈VΓ1&b1,b2∈EΓ2] or [b1=b2∈VΓ2&a1,a2∈EFkΓ1]. For more details, see Figures 2 and 3.Figure 2
(a)Γ1≅P3. (b) Γ2≅P2. (c) Γ1+S2Γ2. (d) Γ1+R2Γ2.
(a)(b)(c)(d)Figure 3
(a)Γ1+Q2Γ2. (b) Γ1+T2Γ2.
(a)(b)Lemma 1.
ForFk∈Sk,Rk,Qk,Tk and x,y∈Γ1+FkΓ2, the degree of x,y in Γ1+FkΓ2 is(5)iddΓ1+SkΓ2x,y=dΓ1x+dΓ2y,ifx∈VΓ1∧y∈VΓ2,2,ifx∈VSkΓ1−VΓ1∧y∈VΓ2,iiddΓ1+RkΓ2x,y=dRkΓ1x+dΓ2y,ifx∈VΓ1∧y∈VΓ2,2,ifx∈VSkΓ1−VΓ1∧y∈VΓ2,iiiddΓ1+QkΓ2x,y=dΓ1x+dΓ2y,ifx∈VΓ1∧y∈VΓ2,dQkΓ1x,ifx∈VQkΓ1−VΓ1∧y∈VΓ2,ivddΓ1+TkΓ2x,y=dTkΓ1x+dΓ2y,ifx∈VΓ1∧y∈VΓ2,dTkΓ1x,ifx∈VTkΓ1−VΓ1∧y∈VΓ2.


## 3. Main Results

The main results of FGZ index of the generalizedF-sum graphs are presented in this section.Theorem 1.
LetΓ1 and Γ2 be two simple graphs and γ∈N−0,1. The FGZ index of the generalized S-sum graph Γ1+SkΓ2 is(6)MγΓ1+SkΓ2=∑i=0ααiMΓ1α−iMΓ2i+1+nΓ2MSkΓ1γ+∑i=1ααiMΓ2iMΓ1γ−i+2α+1k−1nΓ2eΓ1,where N is the set of natural numbers and α=γ−1.Proof.
Let(7)MγΓ1+SkΓ2=∑a,b∈VΓ1+SkΓ2dΓ1+SkΓ2γa,b.
Forα=γ−1, then the above equation is considered as(8)MγΓ1+SkΓ2=∑a,bc,d∈EΓ1+SkΓ2dΓ1+SkΓ2αa,b+dΓ1+SkΓ2αc,d=∑a∈VΓ1∑bd∈EΓ2dΓ1+SkΓ2αa,b+dΓ1+SkΓ2αa,d+∑b∈VΓ2∑ac∈ESkΓ1dΓ1+SkΓ2αa,b+dΓ1+SkΓ2αb,c=∑a∈VΓ1∑bd∈EΓ2dΓ1+SkΓ2αa,b+dΓ1+SkΓ2αa,d+∑b∈VΓ2∑ac∈ESkΓ1a∈VΓ1,c∈VSkΓ1−VΓ1dΓ1+SkΓ2αa,b+dΓ1+SkΓ2αb,c+∑b∈VΓ2∑ac∈ESkΓ1a,c∈VSkΓ1−VΓ1dΓ1+SkΓ2αa,b+dΓ1+SkΓ2αb,c.
For every vertexa∈VΓ1 and edge bd∈EΓ2, then 1st term of (8) will be(9)∑a∈VΓ1∑bd∈EΓ2dΓ1+SkΓ2αa,b+dΓ1+SkΓ2αa,d=∑a∈VΓ1∑bd∈EΓ2∑i=0ααidΓ1α−iadΓ2ib+∑i=0ααidΓ1α−iadΓ2id=∑a∈VΓ1∑bd∈EΓ2∑i=0ααidΓ1α−iadΓ2ib+dΓ2id=∑a∈VΓ1∑i=0ααidΓ1α−ia∑bd∈EΓ2dΓ2ib+dΓ2id=∑a∈VΓ1∑i=0ααidΓ1α−iaMΓ2i+1=∑i=0ααiMΓ1α−iMΓ2i+1.
SinceESkΓ1=2EΓ1. So, for every b∈VΓ2 and ac∈ESkΓ1 with a∈VΓ1, and c∈VSkΓ1−VΓ1; then the 2nd term of (8) is(10)∑b∈VΓ2∑ac∈ESkΓ1a∈VΓ1,c∈VSkΓ1−VΓ1dΓ1+SkΓ2αa,b+dΓ1+SkΓ2αb,c=∑b∈VΓ2∑ac∈ESkΓ1dΓ1+SkΓ2αa,b+dΓ1+SkΓ2αb,c=∑b∈VΓ2∑ac∈ESkΓ1∑i=0ααidSkΓ1α−iadΓ2ib+dSkΓ1αc=∑b∈VΓ2∑ac∈ESkΓ1dSkΓ1αa+∑i=0ααidSkΓ1α−ia.dΓ2ib+dSkΓ1αc=∑b∈VΓ2∑ac∈ESkΓ1dSkΓ1αa+dSkΓ1αc+∑i=1ααidSkΓ1α−iadΓ2ib=∑b∈VΓ2∑ac∈ESkΓ1dSkΓ1αa+dSkΓ1αc+∑b∈VΓ2∑ac∈ESkΓ1∑i=1ααidSkΓ1α−iadΓ2ib=∑b∈VΓ2MSkΓ1α+1+∑i=1ααiMΓ2iMΓ1γ−i=nΓ2MSkΓ1γ+∑i=1ααiMΓ2iMΓ1γ−i,and the 3rd term of equation (8) will be(11)∑bεVΓ2∑acεESkΓ1a,cεVSkΓ1−VΓ1dΓ1+SkΓ2αa,b+dΓ1+SkΓ2αc,b=∑bεVΓ2∑acεESkΓ1a,cεVSkΓ1−VΓ12α+2α.
Since in this caseESkΓ1=k−1|eΓ1, we have(12)=2α+1k−1nΓ2eΓ1.
By using (9), (10), & (12) in (8), we get(13)MγΓ1+SkΓ2=∑i=0ααiMΓ1α−iMΓ2i+1+nΓ2MSkΓ1β+∑i=1ααiMΓ2iMΓ1γ−i+2α+1k−1nΓ2eΓ1.Theorem 2.
LetΓ1 and Γ2 be two simple graphs and γ∈N−0,1. The FGZ index of the generalized R-sum Γ1+RkΓ2 graph is(14)MγΓ1+RkΓ2=∑i=0ααi2α−iMΓ1α−iMΓ2i+1+2∑i=0ααi2α−iMΓ1γ−iMΓ2i+2αeΓ1nΓ2+2α+1k−1nΓ2eΓ1,where N is the set of natural numbers and α=γ−1.Proof.
Then by definition, we have,(15)MγΓ1+RkΓ2=∑a,b∈VΓ1+RkΓ2dΓ1+RkΓ2γa,b.
Forα=γ−1, the above equation is consider as(16)MγΓ1+RkΓ2=∑a,bc,d∈EΓ1+RkΓ2dΓ1+RkΓ2αa,b+dΓ1+RkΓ2αc,d=∑a∈VΓ1∑bd∈EΓ2dΓ1+RkΓ2αa,b+dΓ1+RkΓ2αa,d+∑b∈VΓ2∑ac∈ERkΓ1dΓ1+RkΓ2αa,b+dΓ1+RkΓ2αb,c=∑a∈VΓ1∑bd∈EΓ2dΓ1+RkΓ2αa,b+dΓ1+RkΓ2αa,d+∑b∈VΓ2∑ac∈ERkΓ1a,c∈VΓ1dΓ1+RkΓ2αa,b+dΓ1+RkΓ2αb,c+∑b∈VΓ2∑ac∈ERkΓ1a∈VΓ1c∈VRkΓ1−VΓ1dΓ1+RkΓ2αa,b+dΓ1+RkΓ2αb,c=∑a∈VΓ1∑bd∈EΓ2dΓ1+RkΓ2αa,b+dΓ1+RkΓ2αa,d+∑b∈VΓ2∑ac∈ERkΓ1a,c∈VΓ1dΓ1+RkΓ2αa,b+dΓ1+RkΓ2αb,c+∑b∈VΓ2∑ac∈ERkΓ1a∈VΓ1c∈VRkΓ1−VΓ1dΓ1+RkΓ2αa,b+dΓ1+RkΓ2αb,c∑bεVΓ2∑acεERkΓ1a,cεVRkΓ1−VΓ1dΓ1+RkΓ2αa,b+dΓ1+RkΓ2αc,b.
For every vertexa∈VΓ1& edge bd∈EΓ2, then the 1st term of (16) is(17)∑a∈VΓ1∑bd∈EΓ2dΓ1+RkΓ2αa,b+dΓ1+RkΓ2αa,d=∑a∈VΓ1∑bd∈EΓ2∑i=0ααidRkΓ1α−ia.dΓ2ib+∑i=0ααidRkΓ1α−ia.dΓ2id=∑i=0ααi∑a∈VΓ1dRkΓ1α−ia∑bd∈EΓ2dΓ2ib+∑i=0ααi∑a∈VΓ1dRkΓ1α−ia.∑bd∈EΓ2dΓ2id=∑i=0ααi∑a∈VΓ12dΓ1aα−i∑bd∈EΓ2dΓ2ib+∑i=0ααi∑a∈VΓ12dΓ1aα−i.∑bd∈EΓ2dΓ2id=∑i=0ααi2α−iMΓ1α−i∑bd∈EΓ2dΓ2ib+dΓ2id=∑i=0ααi2α−iMΓ1α−iMΓ2i+1.
For every vertexb∈VΓ2& edge ac∈ERkΓ1a,c∈VΓ1, then the 2nd term of (16) will be(18)∑b∈VΓ2∑ac∈ERkΓ1dΓ1+RkΓ2αa,b+dΓ1+RkΓ2αb,c=∑b∈VΓ2∑ac∈ERkΓ1∑i=0ααidRkΓ1α−ia.dΓ2ib+∑i=0ααidRkΓ1α−ic.dΓ2ib=∑b∈VΓ2∑ac∈ERkΓ1∑i=0ααidΓ2ibdRkΓ1α−ia+dRkΓ1α−ic=∑ac∈ERkΓ1a,c∈VΓ1∑i=0ααi∑b∈VΓ2dΓ2ibdRkΓ1α−ia+dRkΓ1α−ic=∑ac∈ERkΓ1a,c∈VΓ1∑i=0ααiMΓ2i2dΓ1aα−i+2dΓ1cα−i=∑ac∈ERkΓ1a,c∈VΓ1∑i=0ααiMΓ2i2α−idΓ1α−ia+2α−idΓ1α−ic=∑i=0ααiMΓ2i∑ac∈ERkΓ1a,c∈VΓ12α−idΓ1α−ia+2α−idΓ1α−ic=∑i=0ααiMΓ2i∑ac∈ERkΓ1a,c∈VΓ12α−idΓ1α−ia+2α−idΓ1α−ic=∑i=0ααiMΓ2i2α−i∑ac∈ERkΓ1a,c∈VΓ1dΓ1α−ia+dΓ1α−ic=∑i=0ααiMΓ2i2α−1MΓ1γ−i=∑i=0ααiMΓ2i2α−1MΓ1γ−i.
For every vertexb∈VΓ2& edge ac∈ERkΓ1a∈VΓ1,c∈vRkΓ1−VΓ1. Since we have dRkΓ1a=2dΓ1a∀a∈VΓ1 also dRkΓ1c=2∀c∈VRkΓ1−VΓ1. So the 3rd term of (16) will be(19)=∑b∈VΓ2∑ac∈ERkΓ1a∈VΓ1c∈VRkΓ1−VΓ1dΓ1+RkΓ2αa,b+dΓ1+RkΓ2αb,c=∑b∈VΓ2∑ac∈ERkΓ1a∈VΓ1c∈VRkΓ1−VΓ1∑i=0ααidRkΓ1α−ia.dΓ2ib+dRkΓ1αc,
HeredRkΓ1α=2α and dRkΓ1α−ia=2dΓ1aα−i:(20)=∑b∈VΓ2∑ac∈ERkΓ1a∈VΓ1c∈VRkΓ1−VΓ1∑i=0ααi2dΓ1aα−idΓ2ib+2α=∑b∈VΓ2∑ac∈ERkΓ1a∈VΓ1c∈VRkΓ1−VΓ1∑i=0ααi2α−idΓ1aα−idΓ2ib+2α=∑i=0ααi∑b∈VΓ2∑ac∈ERkΓ1a∈VΓ1c∈VRkΓ1−VΓ12α−idΓ1α−iadΓ2ib+∑b∈VΓ2∑ac∈ERkΓ1a∈VΓ1c∈VRkΓ1−VΓ12α=∑i=0ααi2α−i.MΓ2i.MΓ1γ−i+2αeΓ1nΓ2=∑i=0ααi2α−i.MΓ2i.MΓ1γ−i+2αeΓ1nΓ2,and the 4th term of (5) is(21)∑bεVΓ2∑acεERkΓ1a,cεVRkΓ1−VΓ1dΓ1+RkΓ2αa,b+dΓ1+RkΓ2αc,b=∑bεVΓ2∑acεERkΓ1a,cεVRkΓ1−VΓ12α+2α.
Since in this caseESkΓ1=k−1|eΓ1, we have(22)=2α+1k−1nΓ2eΓ1.
Using (17), (18), (20), and (22) in (16), then we have(23)MΓ1+RkΓ2=∑i=0ααi2α−iMΓ1α−iMΓ2i+1+2∑i=0ααi2α−iMΓ1γ−iMΓ2i+2αeΓ1nΓ2+2α+1k−1nΓ2eΓ1.Theorem 3.
LetΓ1 and Γ2 be two simple graphs and γ∈N−0,1. The FGZ index of the generalized Q-sum Γ1+QkΓ2 graph is(24)MγΓ1+QkΓ2=∑i=0ααiMΓ1α−iMΓ2i+1+∑i=0ααiMΓ1γ−iMΓ2i+2nΓ2∑i=0ααidΓ1α−iu.dΓ1iv+nΓ2∑uv∈EΓ1vw∈EΓ1∑i=0ααidΓ1α−iu.dΓ1iv+∑i=0ααidΓ1α−iv.dΓ1iw+2k−1nΓ2∑uvεEΓ1dΓ1αu+dΓ1αv,where N is the set of natural numbers and α=γ−1.Proof.
Then by definition, we have(25)MγΓ1+QkΓ2=∑a,b∈VΓ1+QkΓ2dΓ1+QkΓ2γa,b=∑a,bc,d∈EΓ1+QkΓ2dΓ1+QkΓ2αa,b+dΓ1+QkΓ2αc,d=∑a∈VΓ1∑bd∈EΓ2dΓ1+QkΓ2αa,b+dΓ1+QkΓ2αa,d+∑b∈VΓ2∑ac∈EQkΓ1dΓ1+QkΓ2αa,b+dΓ1+QkΓ2αb,c.
For every vertexa∈VΓ1& edge bd∈EΓ2, then the 1st term of (25) will be(26)∑a∈VΓ1∑bd∈EΓ2dΓ1+QkΓ2αa,b+dΓ1+QkΓ2αa,d=∑a∈VΓ1∑bd∈EΓ2∑i=0ααidQkΓ1α−iadΓ2ib+∑i=0ααidQkΓ1α−iadΓ2id=∑a∈VΓ1∑bd∈EΓ2∑i=0ααidΓ1α−iadΓ2ib+dΓ2id=∑a∈VΓ1∑i=0ααidΓ1α−ia∑bd∈EΓ1dΓ2ib+dΓ2id=∑a∈VΓ1∑i=0ααidΓ1α−iaMΓ2i+1=∑i=0ααiMΓ1α−iMΓ2i+1.
For every vertexb∈VΓ2& edge ac∈EQkΓ1a,c∈VΓ1, then the 2nd term of equation (25) will be(27)∑b∈VΓ2∑ac∈EQkΓ1dΓ1+QkΓ2αa,b+dΓ1+QkΓ2αb,c=∑b∈VΓ2∑ac∈EQkΓ1a∈VΓ1c∈VQkΓ1−VΓ1dΓ1+QkΓ2αa,b+dΓ1+QkΓ2αb,c+∑b∈VΓ2∑ac∈EQkΓ1\\a,c∈VQkΓ1−VΓ1dΓ1+QkΓ2αa,b+dΓ1+QkΓ2αb,c.
Now∀b∈VΓ2, ac∈EQkΓ1 if a∈VΓ1 and c∈VQkΓ1−VΓ1; the 1st term of (27) will be(28)∑b∈VΓ2∑ac∈EQkΓ1a∈VΓ1c∈VQkΓ1−VΓ1dΓ1+QkΓ2αa,b+dΓ1+QkΓ2αb,c=∑b∈VΓ2∑ac∈EQkΓ1a∈VΓ1c∈VQkΓ1−VΓ1dQkΓ1a+dΓ2bα+dQkΓ1αc=∑b∈VΓ2∑ac∈EQkΓ1a∈VΓ1c∈VQkΓ1−VΓ1∑i=0ααidQkΓ1α−ia.dΓ2ib+dQkΓ1αc=∑b∈VΓ2∑ac∈EQkΓ1a∈VΓ1c∈VQkΓ1−VΓ1∑i=0ααidQkΓ1α−ia.dΓ2ib+∑b∈VΓ2∑ac∈EQkΓ1a∈VΓ1c∈VQkΓ1−VΓ1dQkΓ1αc=∑i=0ααi∑b∈VΓ2dΓ2ib∑ac∈EQkΓ1a∈VΓ1c∈VQkΓ1−VΓ1dQkΓ1α−ia+∑b∈VΓ2∑ac∈EQkΓ1a∈VΓ1c∈VQkΓ1−VΓ1dQkΓ1αc=∑i=0ααiMΓ2iMΓ1γ−i+nΓ2.∑ac∈EQkΓ1a∈VΓ1c∈VQkΓ1−VΓ1dQkΓ1αc=∑i=0ααiMΓ2iMΓ1γ−i+nΓ22∑uv∈EΓ1dΓ1u+dΓ1vα=∑i=0ααiMΓ2iMΓ1γ−i+2nΓ2∑i=0ααi∑uv∈EΓ1dΓ1α−iu.dΓ1iv.
Now∀b∈VΓ2& edge ac∈EQkΓ1 if the vertex a,c∈VQkΓ1−VΓ1. Then the 2nd term of equation (27) splits into two parts for the vertices a and c, then the equation will be(29)=∑b∈VΓ2∑ac∈EQkΓ1a,c∈VQkΓ1−VΓ1dΓ1+QkΓ2a,bα+dΓ1+QkΓ2b,cα=∑b∈VΓ2∑ac∈EQkΓ1a,c∈VQkΓ1−VΓ1dΓ1+QkΓ2aα+dΓ1+QkΓ2cα=∑b∈VΓ2∑uv∈EΓ1,vw∈EΓ1dΓ1u+dΓ1vα+dΓ1v+dΓ1wα=cΓ2∑uv∈EΓ1,vw∈EΓ1∑i=0ααidΓ1α−iu⋅dΓ1iv+∑i=0ααidΓ1α−iv.dΓ1iw,(30)∑bεVΓ2∑acεEQkΓ1acεVQkΓ1−VΓ1dQkΓ1αa+dQkΓ1αc=2k−1∑bεVΓ2∑uvεEΓ1dΓ1αu+dΓ1αv=2k−1nΓ2∑uvεEΓ1dΓ1αu+dΓ1αv.
Using (26), (28), (29), and (30) in (25), we get the required result:(31)=∑i=0ααiMΓ1α−iMΓ2i+1+∑i=0ααiMΓ2iMΓ1α+2nΓ2∑i=0ααi∑uv∈E′Γ1dΓ1α−iu.dΓ1iv+nΓ2∑uv∈E′Γ1,vw∈E′Γ1∑i=0ααidΓ1α−iu.dΓ1iv+∑i=0ααidΓ1α−iv.dΓ1iw+2k−1nΓ2∑uvεEΓ1dΓ1αu+dΓ1αv=∑i=0ααiMΓ12γ−2−iMΓ22i+1+2nΓ2∑i=0ααi∑uv∈EΓ1dΓ1α−iu.dΓ1iv+nΓ2∑uv∈EΓ1,vw∈EΓ1∑i=0ααidΓ1α−iu.dΓ1iv+∑i=0ααidΓ1α−iv.dΓ1iw+2k−1nΓ2∑uvεEΓ1dΓ1αu+dΓ1αv.Theorem 4.
LetΓ1 and Γ2 be two simple graphs. The FGZ index of the generalized T-sum graph Γ1+TkΓ2 is(32)MγΓ1+TkΓ2=∑i=0ααi2α−iMΓ1α−iMΓ2i+1+∑i=0ααiMΓ2i2α−iMΓ1γ−i+∑i=0ααi2α−iMΓ2iMΓ1γ−i+2nΓ2∑i=0ααi∑uv∈EΓ1dΓ1α−iu.dΓ1iv+nΓ2∑uv∈EΓ1,vw∈EΓ1∑i=0ααidΓ1α−iu.dΓ1iv+∑i=0ααidΓ1α−iv+dΓ1iw+2k−1nΓ2∑uvϵEΓ1dΓ1αu+dΓ1αv,where γ∈N+−0,1 and α=γ−1.Proof.
Since we havedΓ1+TkΓ2a,b=dΓ1+RkΓ2a,b for every vertex a∈VΓ1 and b∈VΓ2, also dΓ1+TkΓ2a,b=dΓ1+QkΓ2a,b for every vertex a∈VTkΓ1−VΓ1 and b∈VΓ2, the result follows by the proof of Theorems 2 and 3.Theorem 5.
Assume thatΓ1 and Γ2 are two simple graphs and α=γ−1, where γ∈ℜ−0,N+ and ℜ is a set of real number. Then, the FGZ index of generalized F-sum graphs (Γ1+SkΓ2, Γ1+RkΓ2, Γ1+QkΓ2, and Γ1+TkΓ2) are(33)iMγΓ1+SkΓ2=∑i=0∞αiMΓ1iMΓ2α−i+1+nΓ2MSkΓ1i+∑i=1∞αiMΓ2α−1MΓ1i+1+2γ+1k−1nΓ2eΓ1iiMγΓ1+RkΓ2=∑i=0∞αi2iMΓ1iMΓ2α−i+1+2∑i=0∞αi2iMΓ1i+1MΓ2α−i+2γeΓ1nΓ2+2γ+1k−1nΓ2eΓ1iiiMΓ1+QkΓ2γ=∑i=0∞αiMΓ1iMΓ2α−i+1+∑i=0∞αiMΓ1i+1MΓ2α−i+2nΓ2∑i=0∞αidΓ1iu⋅dΓ1α−iv+nΓ2∑uv∈EΓ1,vw∈EΓ1∑i=0∞αidΓ1iu⋅dΓ1α−iv+∑i=0∞αidΓ1iv⋅dΓ1α−iw+2k−1nΓ2∑uvεEΓ1dΓ1γu+dΓ1γvivMγΓ1+TkΓ2=∑i=0∞αi2iMΓ1iMΓ2α−i+1+∑i=0∞αi2iMΓ1i+1MΓ2α−i+∑i=0∞αiMΓ1i+1MΓ2α−i+2nΓ2∑i=0∞αi∑uv∈EΓ1dΓ1iu⋅dΓ1α−iv+nΓ2∑uv∈EΓ1,vw∈EΓ1∑i=0∞αidΓ1iu⋅dΓ1α−iv+∑i=0∞αidΓ1iv⋅dΓ1α−iw+2k−1nΓ2∑uvεEΓ1dΓ1γu+dΓ1γv.Proof.
The above proof is similar as of Theorems1–4.
LetΓ1 be a negative integer, so from Theorem 5, Corollary 1 is obtained.Corollary 1.
Assume thatΓ1&Γ2 are two simple graphs and α=γ−1, where γ is a negative real number. The FGZ index of the generalized F-sums graphs (Γ1+SkΓ2, Γ1+RkΓ2, Γ1+QkΓ2, and Γ1+TkΓ2) are(34)iMγΓ1+SkΓ2=∑i=0∞−1iα+i−1iMΓ1iMΓ2α−i+1+nΓ2MSkΓ1i+∑i=1∞−1iα+i−1iMΓ2α−1MΓ1i+1+2γ+1k−1nΓ2eΓ1iiMγΓ1+RkΓ2=∑i=0∞−1iα+i−1i2iMΓ1iMΓ2α−i+1+2∑i=0∞−1iα+i−1i2iMΓ1i+1MΓ2α−i+2γeΓ1nΓ2+2γ+1k−1nΓ2eΓ1iiiMΓ1+QkΓ2γ=∑i=0∞−1iα+i−1iMΓ1iMΓ2α−i+1+∑i=0∞−1iα+i−1iMΓ1i+1MΓ2α−i+2nΓ2∑i=0∞−1iα+i−1i∑uv∈EΓ1dΓ1iu.dΓ1α−iv+nΓ2∑uv∈EΓ1,vw∈EΓ1∑i=0∞−1iα+i−1idΓ1iu.dΓ1α−iv+∑i=0∞−1iα+i−1idΓ1iv.dΓ1α−iw+2k−1nΓ2∑uvεEΓ1dΓ1γu+dΓ1γvivMγΓ1+TkΓ2=∑i=0∞−1iα+i−1i2iMΓ1iMΓ2α−i+1+∑i=0∞−1iα+i−1i2iMΓ1i+1MΓ2α−i+∑i=0∞−1iα+i−1iMΓ1i+1MΓ2α−i+2nΓ2∑i=0∞−1iα+i−1i∑uv∈EΓ1dΓ1iu⋅dΓ1α−iv+nΓ2∑uv∈EΓ1,vw∈EΓ1∑i=0∞−1iα+i−1idΓ1iu⋅dΓ1α−iv+∑i=0∞−1iα+i−1idΓ1iv⋅dΓ1α−iw+2k−1nΓ2∑uvεEΓ1dΓ1γu+dΓ1γv.


## 4. Applications

Now, we present some examples as applications of the obtained results Theorems1–4. Also the numerical comparisons are represented in Tables 1–4, and the graphical representations are depicted in Figures 4–7.Table 1
Numerical comparison forMγPm+SkPn.
[m,n, k]T = 0T = 1T = 21,1,1−4−4−132,2,22828283,3,31241241334,4,43083083265,5,56046046316,6,61036103610727,7,71628162816738,8,82404240424589,9,933883388345110,10,10460446044676Table 2
Numerical comparison forMγPm+RkPn.
[m,n, k]T = 0T = 1T = 21,1,1−8−2−6.50002,2,23624183,3,316411090.50004,4,44002802355,5,5768558475.0006,6,612929688367,7,7199615341340.58,8,82904228020139,9,9404032302877.510,10,10542844083958Table 3
Numerical comparison forMγPm+QkPn.
[m,n, k]T = 0T = 1T = 21,1,133−1.50002,2,23838473,3,3157157179.50004,4,44084084445,5,5839839888.50006,6,61498149815617,7,7243324332509.58,8,83692369237829,9,9532353235426.510,10,10737473747491Table 4
Numerical comparison forMγPm+TkPn.
[m, n, k]T = 0T = 1T = 21,1,1−3332,2,25442453,3,32271731674,4,45644444175,5,511139038436,6,61922159814937,7,73039257724158,8,84512388836579,9,963895579526710,10,10871876987293Figure 4
Numerical behavior ofMγPm+SkPn using Table 1.Figure 5
Numerical behavior ofMγPm+RkPn using Table 2.Figure 6
Numerical behavior ofMγPm+QkPn using Table 3.Figure 7
Numerical behavior ofMγPm+TkPn.Example 1.
LetPm and Pn be two simple graphs with m≥2 and n≥2. Then, we have(35)1.MγPm+SkPn=∑t=0γCTγ−12γ−1−tm−2+22t+1n−2+2+∑t=0γCTγ−12γ−tm−2+22tn−2+2+n2γ2m−3+2+4k−1nm−1.
From Figure4, it is clear that the behavior of FGZ index of the generalized S-sum graph Γ1+SkΓ2 at t=2 is more better than t=0 and t=1:(36)2.MγPm+RkPn=∑t=0γCTγ−12γ−1−t2γ−1−tm−2+2.2t+1n−2+2+2∑t=0γCTγ−12γ−1−t2γ−tm−2+22tn−2+2+2m−1n+4k−1nm−1.
From Figure5, it is clear that the behavior of FGZ index of the generalized R-sum graph Γ1+RkΓ2 at t=0 is more better than t=1 and t=2:(37)3.MγPm+QkPn=∑t=0γCTγ−12γ−1−tm−2+22t+1n−2+2+∑t=0γCTγ−12γ−tm−2+22tn−2+2+2n∑t=0γCTγ−12γ−1−t+m−22γ−1+2t+n∑t=0γCTγ−12t+2m−12γ−1+2γ−1−t2k−1n2α+1m−2+2.
From Figure6, it is clear that the behavior of FGZ index of the generalized Q-sum graph Γ1+QkΓ2 at t=2 is more better than t=0 and t=1:(38)4.MγPm+TkPn=∑t=0γCTγ−12γ−1−t2γ−1−tm−2+22t+1n−2+2+2∑t=0γCTγ−12γ−1−t2γ−tm−2+22tn−2+2+2n∑t=0γCTγ−12γ−1−t+m−22γ−1+2t+n∑t=0γCTγ−12t+2m−12γ−1+2γ−1−t2k−1n2α+1m−2+2.
From Figure7, it is clear that the behavior of FGZ index of the generalized T-sum graph Γ1+TkΓ2 at t=0 is more better than t=1 and t=2.


## 5. Conclusions

Now, we close our discussion with the following remarks:(i)
For positive integerk and two graphs Γ1&Γ2, we have computed FGZ index of the generalized F-sums graphs Γ1+FkΓ2, where generalized F-sums graphs are obtained by the different operations of subdivision and Cartesian product on Γ1&Γ2.(ii)
The obtained results are also verified and illustrated for the particular classes of graphs.(iii)
The behavior of FGZ index is also analyzed with the help of numerical and graphical presentations.(iv)
However, the problem is still open to compute the different topological indices (degree and distance based) for the generalizedF-sum graphs.



---

*Source: 2954975-2020-12-09.xml*