# Introducing New Exponential Zagreb Indices for Graphs

Authors: Nihat Akgunes; Busra Aydin
Journal: Journal of Mathematics
 (2021)

---

## Abstract

New graph invariants, named exponential Zagreb indices, are introduced for more than one type of Zagreb index. After that, in terms of exponential Zagreb indices, lists on equality results over special graphs are presented as well as some new bounds on unicyclic, acyclic, and general graphs are obtained. Moreover, these new graph invariants are determined for some graph operations.


---

## Body

## 1. Introduction and Preliminaries

LetG be a simple and connected graph with vertex set VG=v1,v2,…,vn and edge set EG such that EG=m. If any two vertices u,v∈VG are adjacent with an edge, they are denoted by uv∈EG. The degree of any vertex v∈VG is the number of edges that are incident to v and denoted by degGv. In this paper, we will generally follow reference [1] for unexplained terminology and notation.In chemical graph theory, different chemical structures are usually modeled by a molecular graph to understand different properties of the chemical compound theoretically. A graph invariant that correlates the physico-chemical properties of a molecular graph with a number is called a molecular structure index. By use of the adjacency, degree, or distance matrices in graph theory, one can describe the structure of molecules in chemistry. The application of molecular structure indices is a standard procedure in structure-property relations, i.e., in QSPR/QSAR studies [2]. As studied in [3], some chemically interesting graphs can be obtained by different graph operations applied onto some general or particular graphs, so it is essential to study such graph operations to understand how they are related to the corresponding topological indices of the original graphs. Some of topological indices are vertex-degree-based, the others are distance or eccentricity-based.The first vertex-degree-based index, called Zagreb, was introduced in [4, 5] under the name of first and second Zagreb indices M1G=∑v∈VGdegv2 and M2G=∑uv∈EGdegudegv, respectively. The first Zagreb index can also be defined as M1G=∑uv∈EGdegu+degv which is used to modeling the best structure-boiling point of alkanes (cf. [6]). Zagreb indices are the oldest and extremely studied graph invariant among others. Readers can refer to the papers, for instance, [6–15] and citations therein. Although there are so many different types of Zagreb indices other than first and second Zagreb indices introduced in the literature, the most common and taken attentions are the first and second multiplicative Zagreb and multiplicative-sum Zagreb indices (see, for instance, [16–19]), Zagreb co-indices (see, for instance, [20]), and finally the multiplicative Zagreb coindices (see, for instance, [8, 9, 20–23]), and citations therein.Later on, Rada [24] has recently introduced a function of exponential topological indices as eφ for a vertex-degree-based index φ and also has studied Randic’s exponential index over the set of graphs with n nonisolated vertices. Furthermore, by using the same approximation eφ, Cruz and Rada [25] have investigated extremal tree graphs for exponential first and second Zagreb indices except maximal tree graph of exponential second Zagreb index. More recently, in [26], the maximal tree graph for exponential second Zagreb index has been obtained. In all these references, the authors defined exponential vertex-degree-based topological indices as the power of exponential.In the light of the thoughts in [24–26], we will first introduce various types of exponential Zagreb indices with a different manner rather than the references in this paragraph. After that, we will investigate some properties over them in separate sections.Definition 1.
For the graphG,(i)
The first and second exponential Zagreb indices are defined by(1)EM1G=∑v∈VGedegv2andEM2G=∑uv∈EGedegudegv.(ii)
The sum version of the first exponential Zagreb index is given by(2)EM1∗G=∑uv∈EGedegu+degv.(iii)
The multiplicative exponential Zagreb indices are defined by(3)EΠ1G=∏v∈VGedegv2andEΠ2G=∏uv∈EGedegudegv,respectively.By considering exponential Zagreb (co)indices given in Definition1 and investigating some new theories via these indices, we aim to point out a new studying field of Zagreb indices for a possible usage in mathematics and mathematical chemistry.The organization of this paper is as follows. We will first present the equalities of the various exponential Zagreb indices obtained by the number of degrees and the number of vertices over well-known graphs. After that, we will show some results on the comparison between exponential Zagreb indices and bounds. Furthermore, we will prove the existence of some new bounds via these new indices over unicyclic and acyclic graphs. Finally, we will state and prove some theorems on graph operations for the new exponential Zagreb indices.


## 2. Results on Special Graphs

In this section, we will present some theorems without proofs since they can be obtained easily by considering the index formulas indicated in Definition1. In detail, we will determine all exponential Zagreb indices for cycle Cn, complete Kn, path Pn, star Sn, wheel Wn, and complete bipartite graphs Km,n, respectively.A direct calculation, by considering the first part of (1) and (2), implies the following theorem.Theorem 1.
The list on the first exponential Zagreb index and the sum version of the first exponential Zagreb index are(4)EM1G=n.e4ifG=Cn,forn≥3,n.en−12ifG=Kn,forn≥3,2e+n−2.e4ifG=Pn,forn≥2,n−1.e+en−12ifG=Sn,forn≥2,n−1.e9+en−12ifG=Wn,forn≥4,m.en2+n.em2ifG=Km,n,form,n≥1,(5)EM1∗G=n.e4ifG=Cn,forn≥3,12n−1.n.e2n−1ifG=Kn,forn≥3,2e3+n−3.e4ifG=Pn,forn≥2,n−1enifG=Sn,forn≥2,n−1.e6+n−1en+2ifG=Wn,forn≥4,m.n.em+nifG=Km,n,form,n≥1,respectively.By the second part of (1), we obtain the second exponential Zagreb index for the same graphs as in the next theorem.Theorem 2.
(6)EM2G=n.e4ifG=Cn,n≥3,12.n−1n.en−12ifG=Kn,forn≥3,2e2+n−3.e4ifG=Pn,forn≥2,n−1.en−1ifG=Sn,forn≥2,n−1.e9+n−1e3n−1ifG=Wn,forn≥4,mn.emnifG=Km,n,form,n≥1.Theorem 3.
By (3), lists on the multiplicative exponential Zagreb indices for some special graphs are as follows:(7)EΠ1G=e4nifG=Cn,forn≥3,enn−12ifG=Kn,forn≥3,e4n−6ifG=Pn,forn≥2,enn−1ifG=Sn,forn≥2,en−1n+8ifG=Wn,forn≥4,emnm+nifG=Km,n,form,n≥1,(8)EΠ2G=e4nifG=Cn,forn≥3,e12nn−13ifG=Kn,forn≥3,e4n−8ifG=Pn,forn≥2,en−12ifG=Sn,forn≥2,e3n−1n+2ifG=Wn,forn≥4,emn2ifG=Km,n,form,n≥1.Theorem 4.
By the first part of (7), lists on the additive exponential Zagreb co-indices are as follows:(9)EM1¯G=n−3n2e4ifG=Cn,forn≥3,0ifG=Kn,forn≥3,e2+2n−3e3+n−4n−32e4ifG=Pn,forn≥2,n−2n−12e2ifG=Sn,forn≥2,n−4n−12e6ifG=Wn,forn≥4,m−1m2e2n+n−1n2e2mifG=Km,n,form,n≥1,(10)EM2¯G=n−3n2e4ifG=Cn,forn≥3,0ifG=Kn,forn≥3,e+2n−3e3+n−4n−32e4ifG=Pn,forn≥2,n−2n−12eifG=Sn,forn≥2,n−4n−12e9ifG=Wn,forn≥4,m−1m2en2+n−1n2em2ifG=Km,n,form,n≥1.Theorem 5.
By the second part of (7), lists on the multiplicative exponential Zagreb coindices are as follows:(11)EΠ1¯G=e2n−3nifG=Cn,forn≥3,0ifG=Kn,forn≥3,e6n+2n−3n−4−16ifG=Pn,forn≥2,en−1n−2ifG=Sn,forn≥2,e3n−1n−4ifG=Wn,forn≥4,em+n−2mnifG=Km,n,form,n≥1,(12)EΠ2¯G=e2n−3nifG=Cn,forn≥3,0ifG=Kn,forn≥3,e4n+2n−3n−4−11ifG=Pn,forn≥2,en−2n−12ifG=Sn,forn≥2,e92n−1n−4ifG=Wn,forn≥4,emn2m−1n+n−1mifG=Km,n,form,n≥1.


## 3. Results on the Comparison between Exponential Zagreb Indices and Some Bounds

The goal of this section is to give some results on the comparison between exponential Zagreb indices and bounds. We need to remind the following facts for our results:Fact 1: it is quite clear that exponential Zagreb index is greater than an ordinary Zagreb index sinceei>i for ∀i>0Fact 2: for any list ofn nonnegative real numbers x1,x2,…,xn, it is well known that the inequality of arithmetic and geometric means (AM-GM inequality) is defined by x1+x2+⋯+xn/n≥nx1x2…xn and the equality holds if and only if x1=x2=⋯=xnTheorem 6.
For a graphG with n vertices and m edges, we have EΠ1G≥e4m2/n.Proof.
By (3) and Cauchy–Schwarz inequality, we obtain(13)EΠ1G=∏u∈VGedegu2=ed12+d22+⋯+dn2=e1/n.d12+d22+⋯+dn2.12+12+⋯+12≥e1/n.d1+d2+⋯+dn2=e1/n.2m2=e4m2/n,as required.Lemma 1.
(see [27]). For any graph G and any edge uv∈EG,(14)∏uv∈EGdegudegv=∏u∈VGdegudegu.Theorem 7.
For a graphG with n vertices and m edges, EΠ2G≥e2m/n2m.Proof.
By equation (3), Fact 2, and Lemma 1, we have(15)EΠ2G=e∑uv∈EGdegudegv=em.∑uv∈EGdegudegvm≥em.∏uv∈EGdegudegvm=em.∏u∈VGdegudegum.
By applyingln into the AM-GM inequality above, we obtain(16)em.ln∑uv∈EGdegudegvm≥em.ln∏u∈VGdegudegum=em.1m.∑i=1ndi.lndiwhere eachdiis the degree of the vertexu∈VG≥e∑i=1ndi.ln∑i=1ndi/n=e∑i=1ndi.ln2m/n=e2m.ln2m/n=2mn2m,which implies(17)∑uv∈EGdegudegvmm≥2mn2m⟶e1m∑uv∈EGdegudegv≥e2mn2.
Henceforth,e∑uv∈EGdegudegv=EΠ2G≥e2m/n2m.We have the following result between two multiplicative exponential Zagreb indices.Theorem 8.
For a simple, connected graphG with n vertices and m edges, let di be the degree of any vertex vi∈VG and Δ be the maximum degree of G. Then, EΠ2G2≤EΠ1GΔ.Proof.
Recall that the first and second Zagreb indices are defined asTex translation failed and Tex translation failed, respectively. Now, for i,j∈1,2,…,n and j≤i, by the fact di2+dj2≤di.di2, we obtain(18)EΠ2G2=e2.M2G=e2∑didj=e∑2didj≤e∑di2+dj2≤e∑di.di2≤e∑Δ.di2=eΔ.M1G=EΠ1GΔ,where each summation defined under the condition i∼j. Hence, the result is obtained.A relation between second exponential Zagreb index and second multiplicative exponential Zagreb index can be given in the next result.Theorem 9.
For any graphG with m edges, it is always true that EM2G/m≥EΠ2Gm.Proof.
SinceEM2G in equation (1) is defined as Tex translation failed, by considering Fact 2, we have(19)EM2Gm=∑i∼jedidjm≥∏i∼jedidjm=EΠ2Gm,which gives the result.There exist the following lower and upper bounds for the sum version of the first exponential Zagreb indexEM1∗G defined in equation (2).Theorem 10.
For a graphG, we have n.e4≤EM1∗G≤n−1n/2e2n−1. Moreover, the equality for lower bound holds if G≃Cn and the equality holds for the upper bound holds if G≃Kn.Proof.
As usual, letdv be the degree of any vertex v∈VG.
It is clear that equation (2) can be notated as Tex translation failed. Since we are interested in the degrees of adjacent vertices in EM1∗G, it is obvious that Kn has the largest degree of all adjacent vertices. Then, we obtain(20)EM1∗Kn=∑u∼uen−1+n−1=n−1n2e2n−1,which implies that Kn is the upper bound for EM1∗G.
Recall that the multiplicative-sum Zagreb index [17] is defined by Tex translation failed, and according to [16], the lower bound for it is either Cn or Pn. Now, by giving attention to the construction of EM1∗G, we easily see that Cn or Pn will be a lower bound. However, EM1∗Pn=2e3+n−3e4>n.e4=EM1∗Cn implies that Cn is the lowest bound for EM1∗G.As we mentioned in Section1, following up [24], Cruz and Rada (Theorem 2.7in [25]) studied about the extremal tree graphs for the exponential first and second Zagreb indices. The result did not cover the maximal tree graph for the exponential of the second Zagreb index. Recently, in Lemma 3 in [26], Zeng and Deng obtained this gap. We should remind that the definition of exponential Zagreb indices in these papers is a different approximation than our way presented in Definition 1.In the following, we will present some results related to the exponential Zagreb and multiplicative exponential Zagreb indices on any simple graphs as well as extremal tree graphs. In fact, proofs of the remaining theorems in this section will be based on the lemmas that will be stated just before results and also based on the truth Fact 1 which was stated at the beginning of this section.Lemma 2.
(see [28]). Among n-vertex tree graphs, the star graph has maximum and the path graph has minimum value of the first Zagreb index. In other words, M1Pn≤M1Tn≤M1Sn.Theorem 11.
For any tree graphTn with n-vertices, we certainly have EM1Pn≤EM1Tn≤EM1Sn and EΠ1Pn≤EΠ1Tn≤EΠ1Sn.The proof of the above theorem can be seen directly by adapting the first parts of equations in (1) and (3) into Lemma 2.Theorem 12.
For any graphG with n-vertices, there exist the orderings EM1G≤EM1Kn and EΠ1G≤EΠ1Kn.Proof.
It is well known that the maximum degree of any vertex in a simple and connected graph withn vertices is n−1. Thus, Tex translation failed.
On the contrary, the product∏ clearly provides the above inequality, and hence, the result is obtained.In [29], Das and Gutman showed that the second Zagreb index M2G decreases when the edges delete from a graph. For this reason, graphs having the greatest and smallest number of edges have maximum and minimum M2G, respectively. With this idea, clearly, the graph with maximum value of second Zagreb index is actually the complete graph as depicted in the next lemma.Lemma 3.
(see [29]). If G has n vertices, then M2Kn¯≤M2G≤M2Kn.Theorem 13.
For any graphG with n vertices, the maximum second exponential Zagreb and the maximum multiplicative second exponential Zagreb indices have orderings:(21)EM2G≤EM2KnandEΠ2G≤EΠ2Kn,respectively.Proof.
Since the second parts of equations (1) and (3) can also be notated as EM2G=∑uv∈EGeM2G and EΠ2G=∏uv∈EGeM2G, respectively, and since eM2G≤eM2Kn holds by Lemma 3, we reach the required orderings.The second Zagreb index version of Lemma2 is as follows.Lemma 4.
(see [29]). Among n-vertex tree graphs, the star graph has the maximum value and the path graph has the minimum value of the second Zagreb index. In other words, M2Pn≤M2Tn≤M2Sn.Now, by adapting the second parts of equations (1) and (3) in Lemma 4, the following result follows immediately.Theorem 14.
For any tree graphTn with n-vertices, we definitely have EM2Pn≤EM2Tn≤EM2Sn and EΠ2Pn≤EΠ2Tn≤EΠ2Sn.


## 4. Results on Unicyclic and Acyclic Graphs

It is known that topological indices are often used for molecular graphs, and unicyclic, bicyclic, and acyclic graphs constitute most of the molecular graphs. In this section, by considering some classes of these graphs, we will state and prove some bounds in terms of first and second exponential Zagreb, sum version of the first exponential Zagreb, and multiplicative exponential Zagreb indices. We remind that results on extremal multiplicative Zagreb indices for trees, unicyclic, and bicyclic graphs are presented, for instance, in [18, 19] and citations therein.We also remind that an acyclic graph is a graph having no graph cycles, and they are actually bipartite as well as a connected acyclic graph is known as a tree. On the contrary, the unicyclic graph is a connected graph containing exactly one cycle. A connected unicyclic graph is, therefore, a pseudotree that is not a tree. We may refer [1] for the details of these terminology.In the following, we will first see the results for some special classes of unicyclic graphs and then see the results for some special classes of acyclic graphs. LetTn,n1,t be a tree of order n with the vertex set VG=v1,v2,…,vn obtained from the path v1v2…vk+n1 by appending the path vk+n1+1…vn to vertex vk for n1≥1 and 2≤k≤n−n1−1. Thus, Un=Tn,n−3,2 is another class of tree graphs. So, these are some classes of acyclic graphs. After all, first and second additive and multiplicative exponential Zagreb indices of above acyclic graphs can be compared as follows.Theorem 15.
For the tree graphsUn and Tn,n1,k with n-vertex, where n1≥1 and 2≤k≤n−n1−1, we have the following equalities and inequalities:(i)
EM1Un=EM1Tn,n1,k and EM2Un<EM2Tn,n1,k(ii)
EM1∗Un<EM1∗Tn,n1,k(iii)
EΠ1Un=EΠ1Tn,n1,k and EΠ2Un<EΠ2Tn,n1,kProof.
Let us considerUn and Tn,n1,k as stated in the theorem.(i)
Letni be the number of the vertices with i degree. Then, all ni’s are equal to each other in Tn,n1,k and Un which implies that EM1Un=EM1Tn,n1,k by the equality in (1). Furthermore, to see the truthfulness of the second inequality, it is enough to apply the second part of (1) into the definitions of Un and Tn,n1,k. Therefore, we have EM2Un=2e2+e3+n−6e4+2e6 and EM2Tn,n1,k=3e2+n−8e4+3e6. Consequently, EM2Un<EM2Tn,n1,k, as required.(ii)
Applying (2) into the definitions of Un and Tn,n1,k, we get EM1∗Un=2e3+n−5e4+2e5 and EM1∗Tn,n1,k=3e3+n−8e4+3e5 that clearly implies the inequality.(iii)
Similarly as ini, by considering the first part of (3), to verify first equality it is enough to see all ni’s are equal to each other in Tn,n1,k and Un where ni is the number of the vertices with i degree. Additionally, for the second inequality, it is sufficient to see that EΠ2Un/EΠ2Tn,n1,k=e2.2+1.3+2.6+4.n−6/e2.3+3.6+4.n−8>1.Since a unicyclic graph has exactly one cycle, the number of vertices and edges are equal. LetUn,k be the set of all unicyclic graphs with n vertices and girth k, where n,k≥3. Suppose also that Ln,k and Hn,k are the subsets of Un,k. In here, we actually have Ln,k=Ck∪Pn−k+1, and the unicyclic graph Hn,k is obtained from Ck by adding n−k pendant vertices to a vertex of Ck.We note thatLn,k=Ck if k=n and Hn,k=Ln,k if k=n−1. So, we suppose that 3≤k≤n−2 in the following theorem.Theorem 16.
For graphsLn,k and Hn,k with n vertices and girth k such that n,k≥3, we have(22)EM1Hn,k≥EM1Ln,kandEΠ1Hn,k≥EΠ1Ln,k.Proof.
Letni be the number of the vertices with degree i. For the graphs Hn,k and Ln,k, these numbers are n1=n−k, n2=2k−n, and n3=n−k and n1=1, n2=n−2, and n3=1, respectively. Consequently,(23)EM1Hn,k−EM1Ln,k=n−ke12+2k−ne22+n−ke32−1.e12+n−2e22+1.e32≥e9+4e4+e≥0,by the assumption 3≤k≤n−2.
To verifyEΠ1Hn,k≥EΠ1Ln,k, it is sufficient to see that(24)EΠ1Ln,kEΠ1Hn,k=e4n+2e6n−2k=e−2≤1,since k≤n−2. Hence, the result is obtained.


## 5. Results on Graph Operations

As mentioned in [30], some chemically interesting graphs can be obtained by different graph operations which are applied onto some general or particular graphs, so it is essential to study such graph operations to understand how they are related to the corresponding topological indices of the original graphs. So far, first and second Zagreb indices have been largely studied under some graph operations (see, for instance, [3, 8, 11, 30–35]).In this section, we will mainly compute the first and second multiplicative exponential Zagreb indices (see equation (3)) in specific graph extensions (or operations) such as Cartesian product G×H, join G+H, lexicographic product GH, tensor product G⊗H, symmetric difference G⊕H, and finally the corona product G∗H for any graphs G and H (we refer any fundamental sources for the definitions of these operations).We have the following known results about these above operations which will be needed for our proofs.Lemma 5.
(see [36]). For any graphs G and H, the following hold:(i)
VG×H=VG⊕H=VG⊗H=VGH=VGVH(ii)
EG×H=EGVH+VGEH(iii)
EG+H=EG+EH+VGVH(iv)
EGH=EGVH2+VGEH(v)
EG⊕H=EGVH2+VG2EH−4EGEH(vi)
G×H is connected if and only if G and H are connectedLemma 6.
(see [37]). For any graphs G and H, the following hold:(1)
Ifa,b∈VG×H, then degG×Ha,b=degGa+degHb.(2)
Ifa,b∈VGH, then degGHa,b=VHdegGa+degHb.(3)
Ifa,b∈VG⊕H, then(25)degG⊕Ha,b=VHdegGa+VGdegHb−2degGadegHb.(4)
Ifa∈VG+H, then degG+Ha=degGa+VHfora∈VGdegHa+VGfora∈VH.Theorem 17.
LetG and H be two graphs. Then, we have(26)EΠ1G×H=Π1GVHΠ1HVGe8EGEHandEΠ2G×H=e3EH.M1G+EGM1H+M2HVG+M2GVH.Proof.
In the following proof, we will consider the facts in Lemma5 (i) and (ii) and Lemma 6 (1).
Now, by adapting the first part of equation (3) in the Cartesian product, we obtain(27)EΠ1G×H=∏u∈VG×HedG×H2u=∏a,b∈VG×HedG×H2a,b=∏a∈VGb∈VHedGa+dHb2=∏a∈VGb∈VHedG2a+dH2b+2dGadHb=∏b∈VH∏a∈VGedG2a∏a∈VG∏b∈VHedH2b∏a∈VG∏b∈VHe2dGadHb=∏b∈VHΠ1G∏a∈VGΠ1H∏a∈VGe2dGa2EH=Π1GVHΠ1HVGe8EGEH.
Similarly, by adapting the second part of equation (3) in the Cartesian product, we obtain(28)EΠ2G×H=∏uv∈EG×HedG×HudG×Hv=∏a,bc,d∈EG×HedG×Ha,bdG×Hc,d=∏a∈VGbd∈EHedG×Ha,bdG×Ha,d.∏ac∈EGb∈VHedG×Ha,bdG×Hc,b=∏a∈VG∏bd∈EHedGa+dHbdGa+dHd.∏ac∈EG∏b∈VHedGa+dHbdGc+dHb=∏a∈VGedG2aEH.∏bd∈EHedGadHb+dHd.edHbdHd.∏b∈VHedH2bEG∏ac∈EGedHbdGa+dGc.edGadGc=∏a∈VGeEH.dG2a.edGaM1H.eM2H.∏b∈VHeEGdH2b.edHbM1G.eM2G=eEH.M1G.e2EGM1H.eM2HVG.eEGM1H.e2EHM1G.eM2GVH=e3EH.M1G+EGM1H+M2HVG+M2GVH.
Hence, the result is obtained.Theorem 18.
For any graphsG and H, let us consider the join graph G+H having vertex and edge sets VG+H=VG∪VH and EG+H=EG∪EH∪xy:x∈VG,y∈VH, respectively. Thus, the first and second multiplicative exponential Zagreb indices are defined by(29)EΠ1G+H=EΠ1G.EΠ1H.eVHVGVH+VG+4VHEG+VGEHEΠ2G+H=EΠ2G.EΠ2H.eVH2VG+EG.eVGM2H.eVHM2G..eEHVG2+4EG+2VH.e2VGEG.Proof.
With the help of Lemma5 (i) and (iii) and Lemma 6 (4) and also adapting the first part of Equation (3) in join graphs, we have(30)EΠ1G+H=∏u∈VG+HedG+H2u=∏u∈VGedG+H2u.∏u∈VHedG+H2u=∏u∈VGedGu+VH2.∏u∈VHedHu+VG2=∏u∈VGedG2u+2dGuVH+VH2.∏u∈VHedH2u+2dHuVG+VG2=EΠ1G.eVH2VG.e4VHEG.EΠ1H.eVG2VH.e4VGEH=EΠ1G.EΠ1H.eVHVGVH+VG+4VHEG+VGEH.
Again considering same lemmas and adapting the second part of Equation (3) in join graphs,(31)EΠ2G+H=∏uv∈EG+HedG+HudG+Hv=∏uv∈EGedG+HudG+Hv∏uv∈EHedG+HudG+Hv∏u∈VGv∈VHedG+HudG+Hv=∏uv∈EGedGu+VHdGv+VH∏uv∈EHedHu+VGdHv+VG∏u∈VGv∈VHedGu+VHdHv+VG=EΠ2G.eVH2EG.eVHM2G.EΠ2H.eVG2EH.eVGM2H.e|VG|VH2.e2VGEG+VHEH.e4EGEH.
This completes the proof.Theorem 19.
LetG and H be graphs. Then, the following holds:(32)EΠ1GH=eVHVH2M1G+8EHEG+VGM1H,EΠ1G⊕H=eVG3M1H.eVH3M1G.∏u∈VGv∈VHe2dGudHvVG−2dGuVH−2dHv,EΠ1G⊗H=eVG3M1H.eVH3M1G.e8VGVHEGEH.eM1HM1G.e−4VHEHM1G.e−4VGEGM1H.Proof.
For simplicity, let us use the following labellings for only in this proof:(33)GH⟶μ,G⊕H⟶η,G⊗H⟶γ,VG⟶v1,EG⟶1,M1G⟶mG,VH⟶v2,EH⟶e2,M1H⟶mH.
Firstly, let us consider the caseμ by taking into account Lemma 5 (i) and (iv) and Lemma 6 (2). So,(34)EΠ1μ=∏u∈Vμedμ2μ=∏a,b∈Vμedμ2a,b=∏a∈VGb∈VHev2dGa+dHb2=∏a∈VGb∈VHev22dG2a+dH2b+2v2dGadHb=∏b∈VHev22mG.∏a∈VGemH.∏a∈VGe2v22e1dGa=ev22v1mG.ev1mH.e2v22e22e1=ev2v22mG+8e2e1+v1mH=EΠ1G.ev22v1.e4v2e1.EΠ1H.ev12v2.e4v1e2=EΠ1G.EΠ1H.ev2v1v2+v1+4v2e1+v1e2.
Now, with the help of Lemma5 (i) and (v) and Lemma 6 (3), let us show truthfulness of the second equality:(35)EΠ1η=∏u,v∈Vηedη2u,v=∏u∈VGv∈VHev1dGv+v2dGu−2dGudHv2=∏u∈VGv∈VHev12dH2v+v22dG2u+2v1v2dGudHv+4dG2udH2v−4dGudHvv1dHv−4dGudHvv2dGu=∏u∈VGev12mH.∏v∈VHev22mG.∏u∈VGv∈VHe2dGudHvv1−2dGuv2−2dHv=ev13mH.ev23mG.∏u∈VGv∈VHe2dGudHvv1−2dGuv2−2dHv.
For the final step,(36)EΠ1γ=∏u,v∈Vγedγ2u,v=∏u∈VGv∈VHev1dGv+v2dGu−dGudHv2=ev13mH.ev23mG.e8v1v2e1e2.emHmG.e−4v2e2mG.e−4v1e1mH,as required.In the following final theorem, we are going to calculate the first and second multiplicative exponential Zagreb indices for the corona product.Theorem 20.
LetG and H be graphs, and let us consider their corona product; then, G∗H. In this case,(37)EΠ1G∗H=eVHVGVH+1+M1G+4VHEG+VGEH+M1HVGEΠ2G∗H=eVH2EG+VG+VGEH+M1H+M2H.e2EG2EH+1+M2G+VHM1G+2VGEH.Proof.
Usually, we will apply Equation (3) into the definition of corona product in our calculations. For simplicity, let us label G∗H by κ:(38)EΠ1κ=∏u∈Vκedκ2u=∏u∈VGedκ2u+∏i=1VG∏u∈VH,iedκ2u=∏u∈VGedGu+VH2+∏i=1VG∏u∈VH,iedGu+12=∏u∈VGedG2u+VH2+2VHdGu+∏i=1VG∏u∈H,iedG2u+1+2dGu=eVH2VG+M1G+4VHEG.eVHVG+M1HVG+4VGEH.
Additionally,(39)EΠ2κ=∏uv∈Eκedκudκv=∏uv∈EGedκudκv.∏i=1VG∏uv∈EHedκudκv.∏uv∈Pedκudκv,where P=uv:u∈VG,v∈VH,i with the index i=1,2,…,VG.
Let us investigate this above last equality in three parts:(40)∏uv∈EGedκudκv=∏uv∈EGedGu+VHdGv+VH=eVH2EG+M2G+VHM1G,(41)∏i=1VG∏uv∈EHedκudκv=∏i=1VG∏uv∈EHedHu+1dHv+1=eVGEH+M1H+M2H,(42)∏uv∈Pedκudκv=∏i=1VG∏u∈VGv∈VHedGu+VHdHv+1=eVH2VG+2VGVHEH+4EGEH+2EG.
Equations (40)–(42) together give the required equality.
Hence, the result is obtained.


## 6. Conclusion and Future Problems

By introducing new graph invariants (see Definition1) under the name of “new generation exponential Zagreb indices,” in here, we obtained some nice equality and inequality results in each section in terms of these new indices. However, there are still some investigations need to be done for future studies. For example, by changing the graphs or graph operations used in this paper, the affection of these new indices can be checked. Additionally, the indices presented in Definition 1 may define another exponential Zagreb co-indices as in the following.For separate verticesu and v, the additive and multiplicative exponential Zagreb coindices are defined by(43)EM1¯G=∑uv∉EGedegu+degv,EM2¯G=∑uv∉EGedegudegv,EΠ1¯G=∏uv∉EGedegu+degv,EΠ2¯G=∏uv∉EGedegudegv.Hence, all results presented and proved in this paper can also be checked with (43).Also, creativity of chemical energy in terms of these new invariants can be determined. Finally, the application of these new indices can be controlled and optimized in QSPR/QSAR studies.



---

*Source: 6675321-2021-05-26.xml*