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Computation of Topological Indices of NEPS of Graphs
The inspection of the networks and graphs through structural properties is a broad research topic with developing significance. One of the methods in analyzing structural properties is obtaining quantitative measures that encode data of the whole network by a real quantity. A large quantity of graph-associated numerical invariants has been used to examine the whole structure of networks. In this analysis, degree-related topological indices have a significant place in nanotechnology and theoretical chemistry. Thereby, the computation of indices is one of the successful branches of research. The noncomplete extended -sum of graphs is a famous general graph product. In this paper, we investigated the exact formulas of general zeroth-order Randić, Randić, and the first multiplicative Zagreb indices for of graphs.
Let be a graph and and be the vertex and edge sets of , respectively. denotes the number of vertices in , which is known as the order of , and the number of elements in is said to be the size of , presented by . The degree of a vertex , presented as , in is the cardinality of linked edges with . Graphs considered in this article are simple, finite, connected, and undirected. Graph theoretical notations which are not interpreted here can be seen in .
Topological invariants have developed a lot of applications in drug discovery, including physicochemical, modeling information of molecules, toxicology, biology, and chemistry. There are numerous graphical invariants, which are effective in chemistry and nanotechnology. Thereby, the calculation of topological invariants is one of the worthwhile domains of graphical research. Suppose that corresponds to the set of all simple graphs and a function is described as a topological invariant if for any set of two isomorphic and graphs, we have . Hundreds of topological indices based on degree or distances have been introduced, but few of them gain popularity due to their high predictive power for various properties such as boiling point, density, refractive index, and molecular weight. For elaborated discussions on these invariants, we refer the reader to [2–11].
Wiener proposed the first distance-related topological invariant, named the Wiener index, and is specified as where is the shortest distance among and in . The authors in [13,14] investigated the Wiener index of several graph operations.
For a graph , Bollobás and Erdos  introduced the general Randić index aswhere . The general Randić index is one of the most eye-catching topological indexes in QSAR and QSPR studies . There is massive research related to the mathematical aspects of the general Randić index [10, 17]. Then, is the classical Randić index . Zeroth-order general Randić index for is a modification of the Randić index [19, 20]. In 2005, Li and Zheng  presented the zeroth-order general Randić index for graphs, which is specified aswhere .
The first and second Zagreb indices are very famous topological descriptors with several implementations in QSPR/QSAR. The first Zagreb index can be obtained by putting in (3), while the second Zagreb index can be achieved by replacing in (2) . The chemical and mathematical aspects of the zeroth-order general Randić index have been investigated in [23–27].
Recently, the authors in  proposed the general multiplicative Zagreb indices for . The first and the second general multiplicative Zagreb indices are described:where .
These are the generalizations of the classic first and second multiplicative Zagreb indices . Khalifeh et al.  investigated the formulas of the Zagreb indices of graph products such as Cartesian product, composition, join, and disjunction. In , the authors investigated the bounds for multiplicative Zagreb indices of graph operations including the join, corona product, Cartesian product, composition, and disjunction.
For graphs and a given set , the noncomplete extended p-sum of graph , , with respect to has the vertex set and the Cartesian product of , , i.e., if is of , then . For , there is an edge if and only if there are some n-tuple such that for , and are distinct, and for . In particular, , and is the graph without any edge on the vertices of .
The operations generalize several known graph products, the Cartesian product of graphs can be acquired by taking , where is an n-dimensional vector in which 1 is placed at the th coordinate and 0 is placed elsewhere, and the tensor product of graphs can be achieved for .
In 2012, Stevanoić  investigated the Zagreb indices for of graphs and showed that inequalities in conjecture unchanged under of graphs. In , the authors determined that the family of gcd-graphs and the family of of complete graphs coincide. For a more detailed study of and related topics, we refer the reader to [34–41]. In this paper, we extended the results of Stevanoić  and investigated the formulas of well-known degree-related topological descriptors for the of graphs.
2. Discussion and Main Results
We start this section with two well-known inequalities which will be beneficial in the proof of the main results.
Lemma 1. Let be nonnegative quantities. Then,and the equality holds if and only if all are equal, .
Lemma 2. Let be nonnegative quantities and be nonnegative weights. Set . If , then we haveand the equality holds if and only if all with are equal, .
Lemma 3 (see ). Let ; then, the degree of a vertex is given as
Next, we derive the expression of the zeroth-order general Randić index of the of graphs for .
Theorem 1. Let , where for , , . If , then
Proof. Let be the graph of . Then, by applying Lemma 3 in (3), we acquirewhich is the required result.
For , the zeroth-order general Randić index gives the famous first Zagreb index, and we have the following corollary.
Corollary 1 (see ). Let , where for , . Then, the first Zagreb index is
Corollary 2. Let be the tensor product of graphs, where with , , and . Then,
We proceed further to describe an upper bound of the zeroth-order general Randić index of graphs for .
Theorem 2. Let , where with , , and . Then,and the equality holds if and only if ’s, , are regular graphs.
Corollary 3. Let be the tensor product of n graphs, where for , , and . Then,and the equality holds if and only if ’s, , are regular graphs.
In the upcoming theorem, we derive the result of graphs with respect to the general Randić index.
Theorem 3. Let , where for , . For , the Randić index of is given aswhere , , and .
Proof. Let ; then, the degree of each vertex is provided by Lemma 3. Thus, the definition of the general Randić index can be described aswhere , , and .
If for some , then from the above result, we obtain the following:In the upcoming result, the general multiplicative Zagreb index of of graphs is discussed.
Theorem 4. Let , where for and . If , thenand the equality holds if and only if s, , are regular graphs.
Proof. Using Lemma 3, the definition of the general multiplicative Zagreb index for can be written asThis completes the proof.
The investigation of graphs with the help of graph invariants is a very successful method, which performs an appreciable part in predicting the physicochemical characterizations of the given graphs. Thereby, the derivations of topological invariants are an attractive, appealing, and interesting line of research in the field of graph theory. The noncomplete extended -sum of graphs is a notable generalization of the graph product. In this paper, we have provided the results of the general zeroth-order Randić, Randić, and first multiplicative Zagreb indices for of graphs. It is more interesting to compute the eccentricity-based indices, PI, and Szeged and Mostar indices of of graphs.
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This research was supported by the UPAR Grants of United Arab Emirates University via Grant nos. G00002590 and G00003271.
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