Research Article  Open Access
Aftab Hussain, Muhammad Numan, Nafisa Naz, Saad Ihsan Butt, Adnan Aslam, Asfand Fahad, "On Topological Indices for New Classes of Benes Network", Journal of Mathematics, vol. 2021, Article ID 6690053, 7 pages, 2021. https://doi.org/10.1155/2021/6690053
On Topological Indices for New Classes of Benes Network
Abstract
Topological indices (TIs) transform a molecular graph into a number. The TIs are a vital tool for quantitative structure activity relationship (QSAR) and quantity structure property relationship (QSPR). In this paper, we constructed two classes of Benes network: horizontal cylindrical Benes network and vertical cylindrical Benes network obtained by identification of vertices of first rows with last row and first column with last column of Benes network, respectively. We derive analytical close formulas for general Randić connectivity index, general Zagreb, first and the second Zagreb (and multiplicative Zagreb), general sum connectivity, atombond connectivity (), and geometric arithmetic index of the two classes of Benes networks. Also, the fourth version of and the fifth version of indices are computed for these classes of networks.
1. Introduction
The processing nodes in an interconnection network are the multiprocessor used to build a network of homogeneously same processor memory pairs. Programs are compiled and executed through message sending. Considerable importance to the architecture and utilization of multiprocessor interconnection network is due to low cost, more efficient microprocessors, and chips [1]. Interconnection networks resembled the communication pattern of the natural scenario, which make it more valuable and important. Most of the networks are interconnected and dependent on each other which need to be reviewed for the future work.
Graphs are used to design interconnected networks in a very natural way, in which the processors or components represent vertices and edges represent the communication links, e.g., fiber optic cables. The way in which all these components work will be carried out by incidence functions. Graphs show the topological properties of the networks. Therefore, the graph and networks are basically the same in a sense; that is, when we are considering networks, components, and links, we actually speak of graphs, vertices, and edges.
In Fourier transform networks, butterfly graphs are the basic graphs that can accomplish FFT very proficiently. There are series of interconnection patterns and switching stages in a butterfly network that permits inputs to be linked to outputs. The Benes network consists of butterflies connected back to back. The Benes network is known for permutation routing [2]. They are significant multistage interconnection networks, which enjoy striking topologies for communication networks [3]. The use of Benes network is in parallel computing systems such as SP1/SP2, IBM, NEC Cenju3, and MIT Transit Project. It is also used in the internal structures of optical couplers [4, 5]. There are levels in an dimensional Benes network where every level has nodes. An dimensional butterfly is formed between the level 0 to level nodes. The back to back butterflies share the middle level to form a Benes network. We denote an dimensional Benes by . Figure 1 shows the graph 3dimensional Benes network .
From now onward, we denote by a connected, simple graph with vertex and edge set by and . Moreover, for , , , and present its degree, neighbors, and sum of degrees. For more details on undefined terminology, we refer the readers to [6–8].
Several invariants assigned to graphs (or molecular structure) establish correlations between its structure and chemical/physical aspects. These graph invariants are named as topological indices. Several degreedependent TIs have been formulated so far and are found to be useful in predicting the physical/chemical properties of an underlying molecular structure. For instance, there is a virtuous correlation among the Randi index (RI) and numerous physicochemical features of alkanes including chromatographic retention times, boiling points, and surface areas. The Zagreb indices (ZIs) and its different variants contribute to understand the complexity of (molecular) structures [9–13], chirality [14], ZE isomerism [15], and heterosystems [16], whilst the overall ZIs show a latent applicability to derive multilinear regression models. The index offers a good model for the strain energy of cycloalkanes and stability of linear and branched alkanes [17, 18]. In this respect, computing different forms of TIs has provided the indicators of such medicinal conduct of numerous compounds and drugs. The mathematical forms of degreebased topological descriptors, which we discuss in this article, are depicted in Table 1. For more details on computation of TIs of different chemical structures, the readers can see [29–36].

In this paper, we constructed two classes of Benes network, i.e., horizontal cylindrical Benes network and vertical cylindrical Benes network obtained by identification of vertices of first rows with last row and first column with last column of Benes network, respectively. Also, we embed the Benes network on torus denoted by . Figures 2 and 3 show the graph of horizontal and vertical Benes networks, respectively. The graph of embedding of Benes network on torus is shown in Figure 4. We compute , , , , , , , , and indices of these networks. Also, the explicit formula of and indices is computed for these networks.
2. Results for Cylindrical Representation of Benes Network
In this section, we construct two types of cylindrical Benes network by identifying its vertices. First, we identify the vertices of top row to the vertices of bottom row of Benes network . This will result in a cylindrical network, and we call it a horizontal cylindrical Benes network and denote it by . Following the construction, a 3dimensional horizontal cylindrical Benes network is presented in Figure 2, where the same label to the vertices represents identification of vertices. Clearly, and , respectively. Now, we compute the , , and of horizontal cylindrical Benes network .
Theorem 1. Let be the graph of horizontal cylindrical Benes network (), then
Proof. In graph , we have vertices among which vertices are of degree 2, 2 vertices are of degree 3, vertices are of degree 4, and the remaining vertices are of degree 6. Now, by using the formula of general Zagreb index, we getTo compute the and indices, we need the partition of the edges of based on the degree of the end vertices. This partition is given in Table 1. Using the values from Table 1, we can compute the and indices as follows:The values of first Zagreb, second Zagreb, and hyperZagreb indices can be computed from the Theorem 6 by taking the value of .
Corollary 1. Let be the graph of horizontal cylindrical Benes network, thenNext, we compute the , , , and indices of horizontal cylindrical Benes network.

Theorem 2. Let be the graph of horizontal cylindrical Benes network, then
Proof. The result follows by using the values of the edge partition given in Table 2 to the formulas of the , , , and indices presented in Table 1.
Now, we compute the values and indices of , where . This requires the partition of its edges based on the sum of degree of neighboring vertices of end vertices of each edge. This partition is given in Table 3.

Theorem 3. Let be the graph of Horizontal cylindrical Benes network , then
Proof. Let denotes the number of edges of with and . Then, by using Table 3, and indices can be computed as follows:
This completes the proof.
Next, we consider the vertical identification of the vertices. We identify the vertices on the left column with the corresponding vertices on the right column to form a cylindrical network. We name this network as the vertical cylindrical Benes network and denote it by . The graph of is shown in Figure 3, where the arcs from vertical sides represent vertical sides identification. The rdimensional vertical cylindrical Benes network has vertices and edges. One can see that is a 4regular graph. Therefore, the computation of TIs for is straightforward and is presented in the following theorem.
Theorem 4. Let be the graph of vertical cylindrical Benes network, then
3. Results for Toroidal Representation of Benes Network
If we identify the vertices in the top row of to the corresponding vertices of the bottom row, we get a toroidal network. We call this network Toroidal Benes network and denote it by . Figure 4 shows the graph of toroidal Benes network . Clearly, and , respectively. Now, we compute the , , and of toroidal Benes network .
Theorem 5. Let be the graph of toroidal cylindrical Benes network, then
Proof. In graph , we have vertices among which vertices are of degree 4 and the remaining vertices are of degree 6. By using the formula of general Zagreb index, we getTo compute the general Randic and general sum connectivity indices, we need the partition of the edges of based on the degree of the end vertices. This partition is given in Table 4. Using the values from Table 4, we can compute the Randic and general sum connectivity indices as follows:The values of first Zagreb, second Zagreb, and hyperZagreb indices can be computed from the Theorem 6 by taking value of and 2.

Corollary 2. Let be the graph of toroidal Benes network, then
Next, we compute the exact values of first multiple Zagreb, second multiple Zagreb, and ABC and GA indices of horizontal cylindrical Benes network.
Theorem 6. Let be the graph of toroidal Benes network, then
Proof. The result follows by using the values of the edge partition given in Table 4 to the formulas of the , , , and indices presented in Table 1.
Now, we compute the values of and indices of , where . This requires the partition of its edges based on the sum of degree of neighboring vertices of end vertices of each edge. This partition is given in Table 5.

Theorem 7. Let be the graph of toroidal Benes network , then
Proof. Let denote the number of edges of with and . Then, by using Table 5, and indices can be computed as follows:This completes the proof.
4. Conclusion and General Remarks
Designing new network structure always attract and open ways for the researchers in the networking and other structural sciences. In this paper, we constructed new classes of Benes networks embedded on the surface of cylinder and torus called horizontal cylindrical Benes network , vertical cylindrical Benes network , and toroidal Benes network . Then, some degreebased TIs are studied for the abovementioned networks. In future, it will help to those who are interested in problems related to interconnected networks and will be able to deal with the complex networks and their topological properties.
Data Availability
No additional data set has been used to support the study.
Disclosure
This research was carried out as a part of the employment of the authors.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
References
 M.S. Chen, K. G. Shin, and D. D. Kandlur, “Addressing, routing, and broadcasting in hexagonal mesh multiprocessors,” IEEE Transactions on Computers, vol. 39, no. 1, pp. 10–18, 1990. View at: Publisher Site  Google Scholar
 V. E. Benes, Mathematical Theory of Connecting Networks and Telephone Traffic, Academic Press, Cambridge, MA, USA, 1965.
 P. D. Manuel, M. I. AbdElBarr, I. Rajasingh, and B. Rajan, “An efficient representation of Benes networks and its applications,” Journal of Discrete Algorithms, vol. 6, no. 1, pp. 11–19, 2008. View at: Publisher Site  Google Scholar
 X. Liu and Q. P. Gu, “Multicasts on WDM alloptical butterfly networks,” Journal of Information Science and Engineering, vol. 18, pp. 1049–1058, 2002. View at: Google Scholar
 J. Xu, Topological Structure and Analysis of Interconnection Networks, Kluwer Academic Publishers, Amsterdam, Netherlands, 2001.
 M. V. Diudea, I. Gutman, and J. Lorentz, Molecular Topology, Nova, Huntington, CA, USA, 2001.
 I. Gutman and O. E. Polansky, Mathematical Concepts in Organic Chemistry, SpringerVerlag, New York, NY, USA, 1986.
 J. Xu, Topological Structure and Analysis of Interconnection Networks, Kluwer Academic, Amsterdam, Netherlands, 2001.
 S. H. Bertz and W. F. Wright, Graph Theory Notes of New York, New York Academy of Sciences, New York, NY, USA, 1998.
 S. Nikolic, I. M. Tolic, and N. Trinajstic, “Relations between three topological indices,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 40, pp. 187–201, 1999. View at: Google Scholar
 S. Nikolic, N. Trinajstic, and I. M. Tolic, “Complexity of molecules,” Journal of Chemical Information and Modeling, vol. 40, pp. 920–926, 2000. View at: Publisher Site  Google Scholar
 S. Nikolic, I. M. Tolic, N. Trinajstic, and I. Bau i, “Complexity of some interesting (chemical) graphs,” Croatica Chemica Acta, vol. 69, no. 3, pp. 883–897, 2000. View at: Google Scholar
 S. Nikolic, N. Trinajstic, I. M. Tolic, G. Rucker, and C. Rucker, “On Molecular Complexity Indices,” in Complexity in Chemistry, D. Bonchev and D. H. Rouvray, Eds., London, UK, 2003. View at: Google Scholar
 M. Ghorbani and M. A. Hosseinzadeh, “Computing ABC4 index of nanostar dendrimers,” Optoelectronics and Advanced MaterialsRapid Communications, vol. 4, pp. 1419–1422, 2010. View at: Google Scholar
 A. Golbraikh, D. Bonchev, and A. Tropsha, “Novel ZEisomerism descriptors derived from molecular topology and their application to QSAR analysis,” Journal of Chemical Information and Computer Sciences, vol. 42, no. 4, pp. 769–787, 2002. View at: Publisher Site  Google Scholar
 A. Milicevic and S. Nikolic, “Graphical matrices in Chemistry,” Croatica Chemica Acta, vol. 77, pp. 97–101, 2004, In press. View at: Google Scholar
 E. Estrada, L. Torres, L. Rodriguez, and I. Gutman, “Modelling the enthalpy of formation of alkanes,” Indian Journal of Chemistry, vol. 37A, pp. 849–855, 1998. View at: Google Scholar
 E. Estrada, “Generalization of topological indices,” Chemical Physics Letters, vol. 336, no. 34, pp. 248–252, 2001. View at: Publisher Site  Google Scholar
 B. Bollobǎs and P. Erdos, “Graphs of extremal weights,” Ars Combinatoria, vol. 50, pp. 225–233, 1998. View at: ǎs &author=P. Erdos&publication_year=1998" target="_blank">Google Scholar
 X. Li and H. Zhao, “Trees with the first three smallest and largest generalized topological indices,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 50, p. 5762, 2004. View at: Google Scholar
 B. Zhou and N. Trinajstić, “On general sumconnectivity index,” Journal of Mathematical Chemistry, vol. 47, no. 1, pp. 210–218, 2010. View at: Publisher Site  Google Scholar
 M. Perc and A. Szolnoki, “Coevolutionary gamesa mini review,” Biosystems, vol. 99, no. 2, pp. 109–125, 2010. View at: Publisher Site  Google Scholar
 M. Randic, “On characterization of molecular branching,” Biosystems, vol. 99, no. 2, pp. 6609–6615, 1975. View at: Publisher Site  Google Scholar
 G. H. Shirdel, H. R. Pour, and A. M. Sayadi, “The hyperZagreb index of graph operations, Iran,” Journal of Mathematical Chemistry, vol. 4, pp. 213–220, 2013. View at: Google Scholar
 M. Ghorbani and N. Azimi, “Note on multiple Zagreb indices, Iran,” Journal of Mathematical Chemistry, vol. 3, no. 2, pp. 137–143, 2012. View at: Google Scholar
 E. Estrada, L. Torres, L. Rodríguez, and I. Gutman, “An atombond connectivity index: modelling the enthalpy of formation of alkanes,” Indian Journal of Chemistry, vol. 37A, pp. 849–855, 1998. View at: Google Scholar
 Z. Du, B. Zhou, and N. Trinajsti, “On geometricarithmetic indices of (molecular) trees, unicyclic graphs and bicyclic graphs,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 66, pp. 681–697, 2011. View at: Google Scholar
 A. Gravovac, M. Ghorbani, and M. A. Hosseinzadeh, “Computing Fth geometricarithmatic index ABC4 index of nanostar dendrimers, Optoelectron,” Advanced Materials–Rapid Communications, vol. 4, no. 9, pp. 1419–1422, 2010. View at: Google Scholar
 K. C. Das and A. Ali, “On a conjecture about the second Zagreb index,” Discrete Mathematics Letters, vol. 2, pp. 38–43, 2019. View at: Google Scholar
 W. Gao, Z. Iqbal, Z. Iqbal, S. Akhter, M. Ishaq, and A. Aslam, “On irregularity descriptors of derived graphs,” AIMS Mathematics, vol. 5, no. 5, pp. 4085–4107, 2020. View at: Publisher Site  Google Scholar
 Z. Iqbal, M. Ishaq, A. Aslam, and W. Gao, “On eccentricity based topological descriptors of water soluble dendrimers,” Zeitschrift für Naturforschung C, vol. 74, no. 12, 2018. View at: Publisher Site  Google Scholar
 A. Aslam, S. Ahmad, M. A. Binyamin, and W. Gao, “Calculating topological indices of certain OTIS interconnection networks,” Open Chemistry, vol. 17, no. 1, pp. 220–228, 2019. View at: Publisher Site  Google Scholar
 A. Aslam, M. F. Nadeem, Z. Zahid, S. Zafar, and W. Gao, “Computing certain topological indices of the line graphs of subdivision graphs of some rooted product graphs,” Mathematics, vol. 7, no. 5, p. 393, 2019. View at: Publisher Site  Google Scholar
 I. Gutman and K. C. Das, “The first Zagreb index 30 years after,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 50, pp. 83–92, 2004. View at: Google Scholar
 K. C. Das, N. Akgunes, M. Togan, A. Yurttas, I. N. Cangul, and A. S. Cevik, “On the first Zagreb index and multiplicative Zagreb coindices of graphs,” Analele Universitatii “Ovidius” ConstantaSeria Matematica, vol. 24, no. 1, pp. 153–176, 2016. View at: Publisher Site  Google Scholar
 N. Akgunes, K. C. Das, and A. S. Cevik, “Topological indices on a graph of monogenic semigroups,” in Topics in Chemical Graph Theory, Mathematical Chemistry Monographs, I. Gutman, Ed., University of Kragujevac and Faculty of Science Kragujevac, Kragujevac, Serbia, 2014. View at: Google Scholar
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Copyright © 2021 Aftab Hussain 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.