Modified Zagreb Connection Indices for Benes Network and Related Classes

The study of networks such as Butterﬂy networks, Benes networks, interconnection networks, David-derived networks through graph theoretical parameters is among the modern trends in the area of graph theory. Among these graph theoretical tools, the topological Indices ( TIs ) have been frequently used as graph invariants. TIs are also the essential tools for quantitative structure activity relationship (QSAR) as well as quantity structure property relationships (QSPR). TIs depend on diﬀerent parameters, such as degree and distance of vertices in graphs. The current work is devoted to the derivation of 2-distance based TIs, known as, modiﬁed ﬁrst Zagreb connection index ZC ∗ 1 and ﬁrst Zagreb connection index ( ZC 1 ) for r − dimensional Benes network and some classes generated from Benes network. The horizontal cylindrical Benes network ( HCB ( r )) , vertical cylindrical Benes network ( VCB ( r )) , and toroidal Benes network ( TB ( r )) are the three classes generated by identifying the vertices of the ﬁrst row with


Introduction
e study of networks such as Butterfly network [1], Benes network [2,3], interconnection network [4][5][6], David-derived network [7] through graph theoretical parameters is among the modern trends in the area of graph theory. In an interconnected network (ICN), the processing nodes are the multiprocessors that are utilized to construct a network based on homogeneously identical processor memory pairs. e transmissions of messages enable programs to be compiled and then executed. Constructive significance to the architectural plan and usage of multiprocessor ICN is on account of economical, reasonable, systematic and more efficient microprocessors and chips [8]. e resemblance of ICN with communication patterns is a natural scenario, which makes them more valuable and influential. Mostly networks are interconnected and due to the dependency on one another, these are networks are needed to be assessed and improved for upcoming work.
To design these networks, graphs are used in a very natural manner, in which the components or processors are distinguished by vertices, and certain communication links such as fiber optic cables etc., are represented by edges. e functionality of the mentioned components is accomplished through incidence functions. It enables to examine networks, its components and the links between components through study of graphs as graphs and networks are similar in the sense of structure.
Butterfly graphs are the elementary graphs in Fourier transform networks that can perform Fast Fourier Transforms (FFT) very expeditiously. In Butterfly networks (BF(r)), the series of interconnection patterns and switching stages permits k inputs to be linked to k outputs. e Benes network comprises back to back connected butterflies.
is worthwhile network is familiar for permutation routing [2]. ese are remarkable multistage interconnection networks, which are entertained by striking and distinctive topologies for communication networks [3]. e Benes networks are utilized in parallel computing systems such as NEC Cenju-3, IBM, SP1/SP2 and MIT Transit Project. ese networks also have applications in the internal structural composition of optical couplers [1,4]. In an r-dimensional Benes, there are 2r + 1 number of levels, with each level having 2r nodes. An r-dimensional butterfly is organized from the level 0 to r nodes. e adjacently connected butterflies which share the central level to generate a Benes network. An r-dimensional Benes network is denoted by B(r), for example, B(3) is shown in Figure 1.
New representations of the Benes networks are recently constructed by embedding it on the surface of torus and cylinder known as Toroidal Benes network (TB(r)), horizontal cylindrical Benes network (HCB(r)), and vertical cylindrical Benes network (VCB(r)). For further details, see [9].
From now onward, G represents a simple, connected graph with edge and vertex sets by E(G) and V(G), respectively. Moreover, for v ∈ V(G), d v and N(v) represent its degree and set of neighbors, respectively. e connection number τ v is the cardinality of the set of vertices which lie at distance 2 from v. For further details on undefined terminologies, we refer [5,10,11]. Various invariants assigned to molecular structures or networks, set up correlations between their physicochemical properties and structures. A class among the graph invariants is the class of topological indices (TIs). ese invariants (TIs) are usually dependent on distance and degree and are came up to be beneficial in anticipating the multiple features of structures including networks and molecular graphs. e first and primordial topological index, entitled as the Wiener index, introduced by H. Wiener [12], in 1947, while studying the alkanes. Based on its productive outcomes and predictive ability, numerous TIs of chemical graphs, have been flourished subsequently. e Zagreb connection index (ZCI) is a noteworthy class of TIs and depends upon connection number denoted by τ v . is connection number expresses the total vertices at distance (edges in a minimal path) two from arbitrary vertex v [13]. is class came into sight in 1972 to quantify the total π-electron energy [14]. After that, researchers took no notice of it for many years. Lately, Ali and Trinajstic [15] reinvestigated the ZCIs and revealed that the ZCI comparatively to classical Zagreb indices come up with finer absolute values of the correlation. Utilizing the connection number ZCI Ali et al. defined the modified first ZCI [15], given as ZC * 1 (G) � v∈V(G) d v τ v and first Zagreb Connection index is defined and denoted as ZC 1 (G) � v∈V(G) τ 2 v [16]. For further details about computation of indices, we refer the readers [17,18]. In this paper, we compute 2-distance based TIs, known as, modified first Zagreb connection index ZC * 1 and first Zagreb connection index (ZC 1 ) for r−dimensional Benes and Butterfly networks, HCB(r), VCB(r), and TB(r). e obtained results are also analyzed through graphical tools.

Main Results
roughout this paper r ≥ 2. Figures 2 and 3 represent the graphs of VCB(r) and HCB(r), respectively. e graph of TB(r), i.e., embedding of B(r) on Torous is given in Figure 4. Now, we present the results in the following sections.

Modified Zagreb Connection Indices for Benes and Butterfly Networks.
In an r−dimensional Benes network B(r), total number of vertices are 2 r (2r + 1), whereas in BF(r), there are 2 r (r + 1) total number of vertices. For details of these networks, see [6]. ese networks are presented in Figures 5 and 6 for r � 3.

Theorem 1.
For r− dimensional Benes network G, we have: be an arbitrary node of Benes network, where i denotes the level (0 ≤ i ≤ 2r) and w � w 1 w 2 . . . w r is an r-bit binary number that denotes the row of the node. Let us denote by V k , 0 ≤ k ≤ 2r, the set of vertices of k th column.
e case when v ∈ V 2r is also the same.

Journal of Mathematics
e case for i � 2r − 1 is similar.

Modified Zagreb Connection Indices for Vertical Cylindrical Representations of Benes Networks.
is network is obtained by the identification of last column with first column of Benes network. Following the construction of VCB(r), clearly there are r2 r+2 vertices in VCB(r). is is a regular graph with degree of each vertex as four [9].

Modified Zagreb Connection Indices for Horizontal Cylindrical Representations of Benes Networks.
e network HCB(r) is obtained by identifying vertices of the last row of Benes network with the corresponding vertices of the first row. e total vertices in HCB(r) are (2r + 1)(2 r − 1) [9]. By following the same method as in previous theorems, the (2r + 1)(2 r − 1) vertices are partitioned in terms of degree and τ as shown in Table 1.  e TB(r) is obtained by the identification of the vertices of bottom row of VCB(r) to the vertices of the top row. Here, benes network is embedded on Torus. e total number of vertices in TB(r) is 2r(2 r − 1) [9]. Now, we compute ZC * 1 and ZC 1 for this network.

Conclusion
e newly generated structures and networks are always interesting topic to be studied. In [9], several new networks such as HCB(r), VCB(r) and TB(r) have been introduced by using B(r). By keeping in view of the importance to study new networks, we computed 2-distance based TIs for these new classes of networks. Moreover, we have also used graphical tools to describe a comparison among the values of the computed indices. Figures 7 and 8 present the rise in the values of the computed TIs of the networks with respect to the size r of the networks B(r), BF(r), VCB(r), HCB(r), and TB(r). e current paper will be a step forward towards the study of these networks for general distance and the maximum distance based descriptors.

Data Availability
No additional data set is used to support the study.