#### 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.

#### 1. Introduction and Preliminaries

Let be a simple and connected graph with vertex set and edge set such that . If any two vertices are adjacent with an edge, they are denoted by . The degree of any vertex is the number of edges that are incident to and denoted by . 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 and , respectively. The first Zagreb index can also be defined as 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 for a vertex-degree-based index and also has studied Randic’s exponential index over the set of graphs with nonisolated vertices. Furthermore, by using the same approximation , 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 graph ,(i)The first and second exponential Zagreb indices are defined by(ii)The sum version of the first exponential Zagreb index is given by(iii)The multiplicative exponential Zagreb indices are defined by respectively.

By considering exponential Zagreb (co)indices given in Definition 1 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 Definition 1. In detail, we will determine all exponential Zagreb indices for cycle , complete , path , star , wheel , and complete bipartite graphs , 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 arerespectively.*

By the second part of (1), we obtain the second exponential Zagreb index for the same graphs as in the next theorem.

Theorem 2.

Theorem 3. *By (3), lists on the multiplicative exponential Zagreb indices for some special graphs are as follows:*

Theorem 4. *By the first part of (7), lists on the additive exponential Zagreb co-indices are as follows:*

Theorem 5. *By the second part of (7), lists on the multiplicative exponential Zagreb coindices are as follows:*

#### 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 since for Fact 2: for any list of nonnegative real numbers , it is well known that the inequality of arithmetic and geometric means (AM-GM inequality) is defined by and the equality holds if and only if

Theorem 6. *For a graph with vertices and edges, we have .*

*Proof. *By (3) and Cauchy–Schwarz inequality, we obtainas required.

Lemma 1. *(see [27]). For any graph and any edge ,*

Theorem 7. *For a graph with vertices and edges, .*

*Proof. *By equation (3), Fact 2, and Lemma 1, we haveBy applying into the AM-GM inequality above, we obtainwhich impliesHenceforth, .

We have the following result between two multiplicative exponential Zagreb indices.

Theorem 8. *For a simple, connected graph with vertices and edges, let be the degree of any vertex and be the maximum degree of . Then, .*

*Proof. *Recall that the first and second Zagreb indices are defined as and , respectively. Now, for and , by the fact , we obtainwhere each summation defined under the condition . 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 graph with edges, it is always true that .*

*Proof. *Since in equation (1) is defined as , by considering Fact 2, we havewhich gives the result.

There exist the following lower and upper bounds for the sum version of the first exponential Zagreb index defined in equation (2).

Theorem 10. *For a graph , we have . Moreover, the equality for lower bound holds if and the equality holds for the upper bound holds if .*

*Proof. *As usual, let be the degree of any vertex .

It is clear that equation (2) can be notated as . Since we are interested in the degrees of adjacent vertices in , it is obvious that has the largest degree of all adjacent vertices. Then, we obtainwhich implies that is the upper bound for .

Recall that the multiplicative-sum Zagreb index [17] is defined by , and according to [16], the lower bound for it is either or . Now, by giving attention to the construction of , we easily see that or will be a lower bound. However, implies that is the lowest bound for .

As we mentioned in Section 1, 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 -vertex tree graphs, the star graph has maximum and the path graph has minimum value of the first Zagreb index. In other words, .*

Theorem 11. *For any tree graph with -vertices, we certainly have and .*

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 graph with -vertices, there exist the orderings and .*

*Proof. *It is well known that the maximum degree of any vertex in a simple and connected graph with vertices is . Thus, .

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 decreases when the edges delete from a graph. For this reason, graphs having the greatest and smallest number of edges have maximum and minimum , 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 has vertices, then .*

Theorem 13. *For any graph with vertices, the maximum second exponential Zagreb and the maximum multiplicative second exponential Zagreb indices have orderings:respectively.*

*Proof. *Since the second parts of equations (1) and (3) can also be notated as and , respectively, and since holds by Lemma 3, we reach the required orderings.

The second Zagreb index version of Lemma 2 is as follows.

Lemma 4. *(see [29]). Among -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, .*

Now, by adapting the second parts of equations (1) and (3) in Lemma 4, the following result follows immediately.

Theorem 14. *For any tree graph with -vertices, we definitely have and .*

#### 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. Let be a tree of order with the vertex set obtained from the path by appending the path to vertex for and . Thus, 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 graphs and with -vertex, where and , we have the following equalities and inequalities:*(i)* and *(ii)*(iii)** and *

*Proof. *Let us consider and as stated in the theorem.(i)Let be the number of the vertices with degree. Then, all ’s are equal to each other in and which implies that 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 and . Therefore, we have and . Consequently, , as required.(ii)Applying (2) into the definitions of and , we get and that clearly implies the inequality.(iii)Similarly as in , by considering the first part of (3), to verify first equality it is enough to see all ’s are equal to each other in and where is the number of the vertices with degree. Additionally, for the second inequality, it is sufficient to see that .

Since a unicyclic graph has exactly one cycle, the number of vertices and edges are equal. Let be the set of all unicyclic graphs with vertices and girth , where . Suppose also that and are the subsets of . In here, we actually have , and the unicyclic graph is obtained from by adding pendant vertices to a vertex of .

We note that if and if . So, we suppose that in the following theorem.

Theorem 16. *For graphs and with vertices and girth such that , we have*

*Proof. *Let be the number of the vertices with degree . For the graphs and , these numbers are , , and and , , and , respectively. Consequently,by the assumption .

To verify , it is sufficient to see thatsince . 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 , join , lexicographic product , tensor product , symmetric difference , and finally the corona product for any graphs and (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 and , the following hold:*(i)*(ii)**(iii)**(iv)**(v)**(vi)** is connected if and only if and are connected*

Lemma 6. *(see [37]). For any graphs and , the following hold:*(1)*If , then .*(2)*If , then .*(3)*If , then*(4)*If , then .*

Theorem 17. *Let and be two graphs. Then, we have*

*Proof. *In the following proof, we will consider the facts in Lemma 5 (i) and (ii) and Lemma 6 (1).

Now, by adapting the first part of equation (3) in the Cartesian product, we obtainSimilarly, by adapting the second part of equation (3) in the Cartesian product, we obtainHence, the result is obtained.

Theorem 18. *For any graphs and , let us consider the join graph having vertex and edge sets and , respectively. Thus, the first and second multiplicative exponential Zagreb indices are defined by*

*Proof. *With the help of Lemma 5 (i) and (iii) and Lemma 6 (4) and also adapting the first part of Equation (3) in join graphs, we haveAgain considering same lemmas and adapting the second part of Equation (3) in join graphs,This completes the proof.

Theorem 19. *Let and be graphs. Then, the following holds:*

*Proof. *For simplicity, let us use the following labellings for only in this proof:Firstly, let us consider the case by taking into account Lemma 5 (i) and (iv) and Lemma 6 (2). So,Now, with the help of Lemma 5 (i) and (v) and Lemma 6 (3), let us show truthfulness of the second equality:For the final step,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. *Let and be graphs, and let us consider their corona product; then, . In this case,*

*Proof. *Usually, we will apply Equation (3) into the definition of corona product in our calculations. For simplicity, let us label by :Additionally,where