Complexity
Volume 2021 (2021), Article ID 9911226, 6 pages
https://doi.org/10.1155/2021/9911226

Research Article

Computation of Topological Indices of NEPS of Graphs

Muhammad Imran,¹ Shehnaz Akhter,² and Muhammad Kamran Jamil³

¹Department of Mathematical Sciences, College of Sciences, United Arab Emirates University, Al Ain, UAE
²School of Natural Sciences, National University of Sciences and Technology, Islamabad, Pakistan
³Department of Mathematics, Riphah Institute of Computing and Applied Sciences, Riphah International University, Lahore, Pakistan

Correspondence should be addressed to Muhammad Imran; imrandhab@gmail.com

Received 3 April 2021; Accepted 2 June 2021; Published 14 June 2021

Academic Editor: Muhammad Javaid

Copyright © 2021 Muhammad Imran et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. Introduction

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 [1].

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 [12]

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 [15] introduced the general Randić index as

where . The general Randić index is one of the most eye-catching topological indexes in QSAR and QSPR studies [16]. There is massive research related to the mathematical aspects of the general Randić index [10, 17]. Then, is the classical Randić index [18]. Zeroth-order general Randić index for is a modification of the Randić index [19, 20]. In 2005, Li and Zheng [21] presented the zeroth-order general Randić index for graphs, which is specified as

where .

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) [22]. The chemical and mathematical aspects of the zeroth-order general Randić index have been investigated in [23–27].

Recently, the authors in [28] 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 [29]. Khalifeh et al. [30] investigated the formulas of the Zagreb indices of graph products such as Cartesian product, composition, join, and disjunction. In [31], 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 [32] investigated the Zagreb indices for of graphs and showed that inequalities in conjecture unchanged under of graphs. In [33], 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 [32] 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 have

and the equality holds if and only if all with are equal, .

Lemma 3 (see [42]). 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 acquire

which 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 [32]). 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.

Proof. For , using Lemma 3 in (3), we have

hence, the required result.

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 as

where , , 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 as

where , , 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 , then

and 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 as

This completes the proof.

3. Conclusion

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.

Data Availability

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.

Acknowledgments

This research was supported by the UPAR Grants of United Arab Emirates University via Grant nos. G00002590 and G00003271.

References

J. A. Bondy and U. S. R. Murty, Graph Theory, Springer, Berlin, Germany, 2008.

S. Akhter and M. Imran, “The sharp bounds on general sum-connectivity index of four operations on graphs,” Journal of Inequalities and Applications, vol. 2016, no. 1, pp. 241–251, 2016.

S. Akhter and M. Imran, “Computing the forgotten topological index of four operations on graphs,” AKCE International Journal of Graphs and Combinatorics, vol. 14, no. 1, pp. 70–79, 2017.

S. Akhter, M. Imran, W. Gao, and R. Farahani, “On topological indices of honeycomb networks and graphene networks,” Hacettepe Journal of Mathematics and Statistics, vol. 47, pp. 19–35, 2018.

S. Akhter, M. Imran, and Z. Raza, “Bounds for the general sum-connectivity index of composite graphs,” Journal of Inequalities and Applications, vol. 76, no. 1, pp. 70–81, 2017.

E. Estrada, The Structure of Complex Networks, Oxford University Press, New York, NY, USA, 2011.

I. Gutman, “Degree-based topological indices,” Croatica Chemica Acta, vol. 86, no. 4, pp. 351–361, 2013.

M. Imran and S. Akhter, “Degree-based topological indices of double graphs and strong double graphs discrete,” Mathematics, Algorithms and Applications, vol. 9, no. 5, Article ID 1750066, 2017.

S. M. Hosamani, R. G. Boli, V. M. Gadag, and V. M. Gadag, “QSPR analysis of certain graph theocratical matrices and their corresponding energy,” Applied Mathematics and Nonlinear Sciences, vol. 2, no. 1, pp. 131–150, 2017.

R. Todeschini and V. Consonni, “New local vertex invariants and molecular descriptors based on functions of the vertex degrees,” MATCH - Communications in Mathematical and in Computer, vol. 64, pp. 359–372, 2010.

H. Yang, M. Imran, S. Akhter, Z. Iqbal, and M. K. Siddiqui, “On distance-based topological descriptors of subdivision vertex-edge join of three graphs,” IEEE Access, vol. 7, pp. 143381–143391, 2019.

H. Wiener, “Structural determination of paraffin boiling points,” Journal of the American Chemical Society, vol. 69, no. 1, pp. 17–20, 1947.

M. Eliasi, G. Raeisi, and B. Taeri, “Wiener index of some graph operations,” Discrete Applied Mathematics, vol. 160, no. 9, pp. 1333–1344, 2012.

Y.-N. Yeh and I. Gutman, “On the sum of all distances in composite graphs,” Discrete Mathematics, vol. 135, no. 1-3, pp. 359–365, 1994.

B. Bollobás and P. Erdös, “Graphs of extremal weights,” Ars Combinatoria, vol. 50, pp. 225–233, 1998.

J. Devillers and A. T. Balaban, Topological Indices and Related Descriptors in QSAR and QSPR, Gordon & Breach, Amsterdam, Netherlands, 1999.

X. Li and I. Gutman, Mathematical Aspects of Randić Type Molecular Structure Description, University of Kragujevac Rectorate, Kragujevac, Serbia, 2006.

M. Randic, “Characterization of molecular branching,” Journal of the American Chemical Society, vol. 97, no. 23, pp. 6609–6615, 1975.

Y. Hu, X. Li, Y. Shi, and T. Xu, “Connected -graphs with minimum and maximum zeroth-order general Randić index,” Discrete Applied Mathematics, vol. 155, pp. 1044–1054, 2007.

H. Hua and H. Deng, “On unicycle graphs with maximum and minimum zeroth-order genenal Randić index,” Journal of Mathematical Chemistry, vol. 41, no. 2, pp. 173–181, 2007.

X. Li and J. Zheng, “A unified approach to the extremal trees for different indices,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 54, pp. 195–208, 2005.

I. Gutman and N. Trinajstić, “Graph theory and molecular orbitals total φ-electron energy of alternant hydrocarbons,” Chemical Physics Letters, vol. 17, no. 4, pp. 535–538, 1972.

A. Ali, A. A. Bhatti, and Z. Raza, “A note on the zeroth-order general randić index of cacti and polyomino chains,” Iranian Journal of Mathematical Chemistry, vol. 5, pp. 143–152, 2014.

H. Ahmed, A. A. Bhatti, and A. Ali, “Zeroth-order general Randić index of cactus graphs,” AKCE International Journal of Graphs and Combinatorics, vol. 16, no. 2, pp. 182–189, 2019.

Y. Hu, X. Li, Y. Shi, T. Xu, and I. Gutman, “On molecular graphs with smallest and largest zeroth-order general randić index,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 54, no. 2, pp. 425–434, 2005.

S. Li and M. Zhang, “Sharp bounds on the zeroth-order general randić indices of conjugated bicyclic graphs,” Mathematical and Computer Modelling, vol. 53, no. 9-10, pp. 1990–2004, 2011.

G. Su, L. Xiong, X. Su, and G. Li, “Maximally edge-connected graphs and zeroth-order general Randić index for $$\alpha \le -1$$ α ≤ −1,” Journal of Combinatorial Optimization, vol. 31, no. 1, pp. 182–195, 2016.

T. Vetrík and S. Balachandran, “General multiplicative Zagreb indices of trees,” Discrete Applied Mathematics, vol. 247, pp. 341–351, 2018.

R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley VCH, Weinheim, Germany, 2000.

M. H. Khalifeh, H. Yousefi-Azari, and A. R. Ashrafi, “The first and second Zagreb indices of some graph operations,” Discrete Applied Mathematics, vol. 157, no. 4, pp. 804–811, 2009.

K. C. Das, A. Yurttas, M. Togan, A. Cevik, and I. Cangul, “The multiplicative zagreb indices of graph operations,” Journal of Inequalities and Applications, vol. 2013, no. 1, p. 90, 2013.

D. Stevanoić, “Comparing the zagreb indices of the NEPS of graphs,” Applied Mathematics and Computation, vol. 219, pp. 1082–1086, 2012.

W. Klotz and T. Sander, “GCD-graphs and NEPS of complete graphs,” Ars Mathematica Contemporanea, vol. 6, pp. 289–299, 2013.

M. Berhe Belay and C. Wang, “The first general Zagreb coindex of graph operations,” Applied Mathematics and Nonlinear Sciences, vol. 5, no. 2, pp. 109–120, 2020.

J.-B. Liu, J. Zhao, H. He, and Z. Shao, “Valency-based topological descriptors and structural property of the generalized sierpiński networks,” Journal of Statistical Physics, vol. 177, no. 6, pp. 1131–1147, 2019.

J.-B. Liu, J. Zhao, and Z.-Q. Cai, “On the generalized adjacency, laplacian and signless laplacian spectra of the weighted edge corona networks,” Physica A: Statistical Mechanics and Its Applications, vol. 540, Article ID 123073, 2020.

J.-B. Liu and X.-F. Pan, “Minimizing kirchhoff index among graphs with a given vertex bipartiteness,” Applied Mathematics and Computation, vol. 291, pp. 84–88, 2016.

D. Stevanović, “When is NEPS of graphs connected?” Linear Algebra and Its Applications, vol. 301, pp. 137–144, 1999.

D. Stevanović, “When can the components of NEPS be almost cospectral?” Linear Algebra and Its Applications, vol. 311, pp. 35–44, 2000.

D. Stevanović, “On the components of NEPS of connected bipartite graphs,” Linear Algebra and Its Applications, vol. 356, pp. 67–78, 2002.

D. Stevanoviz, “Energy and NEPS of graphs,” Linear and Multilinear Algebra, vol. 53, no. 1, pp. 67–74, 2005.

D. Cvetković, M. Doob, and H. Sachs, Spectra of Graphs Theory and Application, Johann Ambrosius Barth Verlag, Heidelberg, Leipzig, 1995.