# Modified Zagreb Connection Indices for Benes Network and Related Classes

Authors: Wenhu Wang; Asma Nisar; Asfand Fahad; Muhammad Imran Qureshi; Abdu Alameri
Journal: Journal of Mathematics
 (2022)

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## Abstract

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 IndicesTIs 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 ZC1∗ and first Zagreb connection index ZC1 for r− dimensional Benes network and some classes generated from Benes network. The horizontal cylindrical Benes network HCBr, vertical cylindrical Benes network VCBr, and toroidal Benes network TBr are the three classes generated by identifying the vertices of the first row with the last row, the first column with the last column of the Benes network. The obtained results are also analyzed through graphical tools.


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## Body

## 1. Introduction

The study of networks such as Butterfly network [1], Benes network [2, 3], interconnection network [4–6], 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. The 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]. The 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. The 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 TransformsFFT very expeditiously. In Butterfly networks BFr, the series of interconnection patterns and switching stages permits k inputs to be linked to k outputs. The Benes network comprises back to back connected butterflies. This worthwhile network is familiar for permutation routing [2]. These are remarkable multistage interconnection networks, which are entertained by striking and distinctive topologies for communication networks [3]. The Benes networks are utilized in parallel computing systems such as NEC Cenju-3, IBM, SP1/SP2 and MIT Transit Project. These 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. The adjacently connected butterflies which share the central level to generate a Benes network. An r-dimensional Benes network is denoted by Br, for example, B3 is shown in Figure 1.Figure 1
3-dimensional Benes network.New representations of the Benes networks are recently constructed by embedding it on the surface of torus and cylinder known as Toroidal Benes networkTBr, horizontal cylindrical Benes network HCBr, and vertical cylindrical Benes network VCBr. For further details, see [9].From now onward,G represents a simple, connected graph with edge and vertex sets by EG and VG, respectively. Moreover, for v∈VG, dv and Nv represent its degree and set of neighbors, respectively. The connection number τv 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. These 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. The 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. The Zagreb connection index ZCI is a noteworthy class of TIs and depends upon connection number denoted by τv. This connection number expresses the total vertices at distance (edges in a minimal path) two from arbitrary vertex v [13]. This 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 ZC1∗G=∑v∈VGdvτv and first Zagreb Connection index is defined and denoted as ZC1G=∑v∈VGτv2 [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 ZC1∗ and first Zagreb connection index ZC1 for r−dimensional Benes and Butterfly networks, HCBr, VCBr, and TBr. The obtained results are also analyzed through graphical tools.


## 2. Main Results

Throughout this paperr≥2. Figures 2 and 3 represent the graphs of VCBr and HCBr, respectively. The graph of TBr, i.e., embedding of Br on Torous is given in Figure 4. Now, we present the results in the following sections.Figure 2
Normal representation ofVCB3.Figure 3
Normal representation ofHCB3.Figure 4
Normal representation ofTB3.


### 2.1. Modified Zagreb Connection Indices for Benes and Butterfly Networks

In anr−dimensional Benes network Br, total number of vertices are 2r2r+1, whereas in BFr, there are 2rr+1 total number of vertices. For details of these networks, see [6]. These networks are presented in Figures 5 and 6 for r=3.Figure 5
Normal representation of B (3).Figure 6
Normal representation of BF (3).Theorem 1.
Forr−dimensional Benes networkG, we have:(1)
ZC1∗G=2r+242+20r−3(2)
ZC1G=200r.2r−269.2rPROOF.
Letw,i=w1w2…wr,i be an arbitrary node of Benes network, where i denotes the level 0≤i≤2r and w=w1w2…wr is an r-bit binary number that denotes the row of the node. Let us denote by Vk, 0≤k≤2r, the set of vertices of k th column. For 0≤i,j≤2r, if w,i and w′,j are adjacent then wj=0 implies wj′=1, and vice versa. We partition the vertices v=w,i, 0≤i≤2r of benes networks into the different cases depending on τv:Case 1.
Fori=0andi=2r
Letv∈V0, that is v=w1w2…wr,0. Then Nv=Nw1…wr,0=w1…wr,1,w1′…wr,1. Furthermore, Nw1w2…wr,1=w1…wr,0,w1…wr,2,w1′…wr,0,w1w2′…wr,2 and.Nw1′…wr,1=w1…wr,0,w1′…wr,2,w1′…wr,0,w1′w2′…wr,2. So, there are 5 distinct vertices that are at distance 2 from v, that is τv=5 for v∈V0. On the other hand for v∈V0, dv=2. Thus, ∑v∈V0dvτv=2.5.2r=10.2r. The case when v∈V2r is also the same.Case 2.
Fori=r
Letv∈Vr, that is v=w1…wr,r. Then Nv=w1…wr,r+1,w1…wr′,r+1,w1…wr,r−1,w1…wr′,r−1. Furthermore, Nw1…wr,r+1=w1…wr,r,w1…wr,r+2,w1…wr−1′wr,r+2,w1…wr′,r, Nw1…wr′,r+1=w1…wr,r,w1…wr′,r+2,w1…wr−1′wr′,r+2,w1…wr′,r, Nw1…wr,r−1=w1…wr,r,w1…wr,r−2,w1…wr−1′wr,r−2,w1…wr′,r and Nw1…wr′,r−1=w1…wr,r,w1…wr′,r−2,w1…wr−1′wr′,r−2,w1…wr′,r.
Thus,τv=9 for v∈Vr. On the other hand for v∈Vr, dv=4. So, ∑v∈Vrdvτv=4.9.2r=36.2r.Case 3.
Fori=1,i=2r−1
Letv∈V1, that is v=w1w2…wr,1. Then Nv=w1…wr,2,w1w2′…wr,2,w1…wr,0,w1′…wr,0. Furthermore, Nw1…wr,2=w1…wr,1,w1…wr,3,w1w2′…wr,1,w1w2w3′…wr,3, Nw1w2′…wr,2=w1…wr,1,w1w2′…wr,3,w1w2′…wr,1,w1w2′w3′…wr,3, Nw1w2…wr,0=w1w2…wr,1,w1′w2…wr,1 and.
Nw1′…wr,0=w1w2…wr,1,w1′w2…wr,1. Thus, τv=6 for v∈V1. On the other hand for v∈V1, dv=4. So, ∑v∈V1dvτv=4.6.2r=24.2r. The case for i=2r−1 is similar.Case 4.
Fori=r−1,i=r+1
Letv∈Vr−1, that is v=w1…wr,r−1. Then Nv=w1…wr,r,w1…wr′,r,w1…wr,r−2,w1…wr−1′wr,r−2. Furthermore, Nw1…wr,r=w1…wr,r−1,w1…wr,r+1,w1…wr′,r+1,w1…wr′,r−1, Nw1…wr′,r=w1…wr,r−1,w1…wr′,r+1,w1…wr′,r−1,w1…wr,r+1, Nw1…wr,r−2=w1…wr,r−1,w1…wr−2′wr−1wr,r−3,w1…wr,r−3,w1…wr−1′wr,r−1 and Nw1…wr−1′wr,r−2=w1…wr,r−1,w1…wr−1′wr,r−1,w1…wr−2′wr−1′wr,r−3,w1…wr−1′wr,r−3.
Thusτv=8 for v∈Vr−1. On the other hand for v∈Vr−1, dv=4. So ∑v∈Vr−1dvτv=4.8.2r=32.2r. The case for i=r+1 is similar.Case 5.
For1<i<r−1,r+1<i<2r−1
Letv∈Vi, where 1<i<r−1 or r+1<i<2r−1, that is v=w1…wr,i. Then Nv=w1…wr,i+1,w1…wi+1′…wr,i+1,w1…wr,i−1,w1…wi′…wr,i−1. Furthermore, Nw1…wr,i+1=w1…wr,i,w1…wr,i+2,w1…wi+2′…wr,i+2,w1…wi+1′…wr,i, Nw1…wr,i−1=w1…wr,i,w1…wr,i−2,w1…wi−1′…wr,i−2,w1…wi′…wr,i, Nw1…wi+1′…wr,i+1=w1…wr,i,w1…wi+1′…wr,i+2,w1…wi+1′…wr,i,w1…wi+1′wi+2′…wr,i+2 and Nw1…wi′…wr,i−1=w1…wr,i,w1…wi′…wr,i−2,w1…wi′…wr,i,w1…wi−1′wi′…wr,i−2. Thus τv=10, on the other hand dv=4 for v∈Vi. So ∑v∈Vidvτv=4.10.2rr−3=40.2rr−3. Hence ZC1∗G=∑v∈VGdvτv=2.10.2r+36.2r+2.32.2r+2.24.2r+2.40r−32r=2r+242+20r−3.
Also,ZC1G=2.25.2r+81.2r+2.36.2r+2.64.2r+2.100.2rr−3=200r.2r−269.2r.Theorem 2.
Forr−dimensional Butterfly networkG, we have(1)
ZC1∗G=2r40r−52(2)
ZC1G=100r.2r−178.2rPROOF.
Letv∈Vr in a butterfly network G, that is v=w1…wr,r. Then Nv=w1…wr,r−1,w1…wr′,r−1. Furthermore, Nw1…wr,r−1=w1…wr′,r,w1…wr,r,w1…wr,r−2,w1…wr−1′wr,r−2 and
Nw1…wr′,r−1=w1…wr−1′wr′,r−2],[w1…wr′,r][[w1…wr′,r−2],[w1…wr,r. Thus, for v∈Vr, τv=5 and dv=2. The case when v∈V0 is also same.
Moreover, ifv∈Vr−1 that is v=w1w2…wr,r−1. Then Nv=w1…wr,r,w1…wr−1′wr,r−2,w1…wr,r−2,w1…wr′,r and Nw1…wr−1′wr,r−2=w1…wr,r−1,w1…wr−1′wr,r−1,w1…wr−1′wr,r−3w1…wr−2′wr−1′wr,r−3Nw1…wr,r−2=w1…wr,r−1,w1…wr−1′wr,r−1,w1…wr−2′wr−1wr,r−3,w1…wr,r−3, Nw1w2…wr,r=w1w2…r,r−1,w1…wr′,r−1 and. Nw1…wr′,r=w1w2…wr,r−1,w1…wr′,r−1.
Thus, forv∈Vr−1, τv=6 and dv=4. The case for v∈V1 is also similar.
By following the same pattern, it can be proved that ifv∈Vi for 2≤i≤r−2, dv=4 and τv=10. So, ZC1∗G=2.5.2r+4.6.2r+4.10.2rr−3+2.5.2r+4.6.2r=2r40r−52 and ZC1G=25.2r+36.2r+100.2rr−3+36.2r+25.2r=100r.2r−178.2r.


### 2.2. Modified Zagreb Connection Indices for Vertical Cylindrical Representations of Benes Networks

This network is obtained by the identification of last column with first column of Benes network. Following the construction ofVCBr, clearly there are r2r+2 vertices in VCBr. This is a regular graph with degree of each vertex as four [9].Theorem 3.
ForG=VCBr, we have(1)
ZC1∗G=2r200+80r−3(2)
ZC1G=200r.2r−182.2rPROOF.
LetG denotes the 4-regular graph of VCBr in which w,i=w1w2…wr,i is an arbitrary node, for any 0≤i≤2r−1. Let Vk, 0≤k≤2r−1 be the set of vertices of k th column in G. For 0≤i,j≤2r−1, if w,i and w′,j are adjacent then wj=0 implies wj′=1, and vice versa. We partition the vertices v=w,i, 0≤i≤2r−1 of G in to the different cases depending on τv:Case 6.
Fori=0
Letv∈V0, that is v=w1w2…wr,0, then Nv=Nw1…wr,0=w1…wr,1,w1′…wr,1,w1…wr,2r−1,w1′…wr,2r−1.
Furthermore,Nw1w2…wr,1=w1…wr,0,w1…wr,2,w1′…wr,0,w1w2′…wr,2Nw1′…wr,1=w1…wr,0,w1′…wr,2,w1′…wr,0,w1′w2′…wr,2, Nw1…wr,2r−1=w1…wr,2r−2,w1w2′…wr,2r−2,w1′…wr,0,w1…wr,0 and. Nw1′…wr,2r−1=w1′…wr,2r−2,w1′…wr,0,w1′w2′…wr,2r−1,w1…wr,0. So, τv=9 and therefore, ∑v∈V0dvτv=4.9.2r=36.2r.Case 7.
Fori=1,i=2r−1
Letv∈V1, that is v=w1w2…wr,1. Then Nv=w1…wr,2,w1w2′…wr,2,w1…wr,0,w1′…wr,0.
Furthermore,Nw1w2…wr,2=w1…wr,1,w1…wr,3,w1w2w3′…wr,3,w1w2′…wr,1, Nw1w2′…wr,2=w1…wr,1,w1w2′…wr,3,w1w2′…wr,1,w1w2′w3′…wr,3,
Nw1…wr,0=w1…wr,2r−1],[w1′…wr,2r−1][[w1′…wr,1],[w1…wr,1 and Nw1′…wr,0=w1′…wr,2r−1,w1…wr,2r−1,w1′w2′…wr,2r−1,w1…wr,1. So, τv=8 and therefore ∑v∈V1dvτv=4.8.2r=32.2r. The case for i=2r−1 is similar.Case 8.
For1<i<2r−1
Similarly, ifv∈Vi, where 1<i<2r−1, then we have ∑v∈Vi=2.40r−32r+2.32.2r+36.2r=2r80r−3+100. Therefore, ZC1∗G=∑v∈VGdvτv=36.2r+2.32.2r+2r80r−3+100=2r200+80r−3 and ZC1G=81.2r+2.64.2r+2.100r−3.2r+2.64.2r+81.2r=200r.2r−182.2r.


### 2.3. Modified Zagreb Connection Indices for Horizontal Cylindrical Representations of Benes Networks

The networkHCBr is obtained by identifying vertices of the last row of Benes network with the corresponding vertices of the first row. The total vertices in HCBr are 2r+12r−1 [9]. By following the same method as in previous theorems, the 2r+12r−1 vertices are partitioned in terms of degree and τ as shown in Table 1.Table 1
Vertex partition ofHCBr corresponding to connection numbers and degrees.
DegτvVertices3926112614261616182 (r − 3)252 (2r − 34)272 (2)462 (2r − 36)482 (2)472 (2)4822r−64102 (4)4122492r−441022r−4 (r − 33)4142 (2)Theorem 4.
ForG=HCBr, we have(1)
ZC1∗G=802+104r−72.2r+80r.2r(2)
ZC1G=200r.2r−269.2r−152r+2304PROOF.
From the partition of the vertices ofHCBr and Table 1 comprising connection number of the vertices, we have
ZC1∗G=∑v∈VGdvτv=54+132+168+96+216r−3+202r−4+56+482r−6+128+112+642r−6+320+96+362r−4+802r−4r−3+224=1386+216r−648+20.2r−80+48.2r−288+64.2r−384+36.2r−144+80r.2r−r320−240.2r+960=802−104r−72.2r+80r.2r.
Similarly,ZC1G=162+242+392+256+648r−1944+50.2r−200+196+72.2r−432+256+196+128.2r−768+800+288+81.2r−324+200.r.2r−800r−600.2r+2400+784=200r.2r−269.2r−152r+2304.


### 2.4. Modified Zagreb Connection Indices for Toroidal Representation of Benes Networks

TheTBr is obtained by the identification of the vertices of bottom row of VCBr to the vertices of the top row. Here, benes network is embedded on Torus. The total number of vertices in TBr is 2r2r−1 [9]. Now, we compute ZC1∗ and ZC1 for this network.Theorem 5.
ForG=TBr, we have(1)
ZC1∗G=80r.2r−40.2r−104r+1044(2)
ZC1G=200r.2r−182.2r−152r+3020PROOF.
The2r2r−1 vertices of TBr are partitioned on the basis of degree as: 2r2r−2 and 2r vertices have degrees 4 and 6, respectively. The connection numbers of vertices of TBr are presented in Table 2.
So, we have
ZC1∗G=∑v∈VGdvτv=108+168+168+96+216r−3+362r−4+192+642r−4+176+112+642r−6+320+96+362r−4+802r−4r−3+224=80r.2r−40.2r−104r+1044.
Moreover,ZC1G=324+392+392+256+648r−1944+81.2r−324+576+128.2r−512+484+196+128.2r−768+800+288+81.2r−324+200.r.2r−800r−600.2r+2400+784=200r.2r−182.2r−152r+3020.Table 2
Vertex partition ofTBr corresponding to connection numbers and degree.
DegτvNo. of vertices61816142614261616182 (r-3)49(2r-4)4122 (2)482 (2r-4)4112 (2)472 (2)4822r−64102 (4)4122492r−441022r−4 (r-3)4142 (2)


## 2.1. Modified Zagreb Connection Indices for Benes and Butterfly Networks

In anr−dimensional Benes network Br, total number of vertices are 2r2r+1, whereas in BFr, there are 2rr+1 total number of vertices. For details of these networks, see [6]. These networks are presented in Figures 5 and 6 for r=3.Figure 5
Normal representation of B (3).Figure 6
Normal representation of BF (3).Theorem 1.
Forr−dimensional Benes networkG, we have:(1)
ZC1∗G=2r+242+20r−3(2)
ZC1G=200r.2r−269.2rPROOF.
Letw,i=w1w2…wr,i be an arbitrary node of Benes network, where i denotes the level 0≤i≤2r and w=w1w2…wr is an r-bit binary number that denotes the row of the node. Let us denote by Vk, 0≤k≤2r, the set of vertices of k th column. For 0≤i,j≤2r, if w,i and w′,j are adjacent then wj=0 implies wj′=1, and vice versa. We partition the vertices v=w,i, 0≤i≤2r of benes networks into the different cases depending on τv:Case 1.
Fori=0andi=2r
Letv∈V0, that is v=w1w2…wr,0. Then Nv=Nw1…wr,0=w1…wr,1,w1′…wr,1. Furthermore, Nw1w2…wr,1=w1…wr,0,w1…wr,2,w1′…wr,0,w1w2′…wr,2 and.Nw1′…wr,1=w1…wr,0,w1′…wr,2,w1′…wr,0,w1′w2′…wr,2. So, there are 5 distinct vertices that are at distance 2 from v, that is τv=5 for v∈V0. On the other hand for v∈V0, dv=2. Thus, ∑v∈V0dvτv=2.5.2r=10.2r. The case when v∈V2r is also the same.Case 2.
Fori=r
Letv∈Vr, that is v=w1…wr,r. Then Nv=w1…wr,r+1,w1…wr′,r+1,w1…wr,r−1,w1…wr′,r−1. Furthermore, Nw1…wr,r+1=w1…wr,r,w1…wr,r+2,w1…wr−1′wr,r+2,w1…wr′,r, Nw1…wr′,r+1=w1…wr,r,w1…wr′,r+2,w1…wr−1′wr′,r+2,w1…wr′,r, Nw1…wr,r−1=w1…wr,r,w1…wr,r−2,w1…wr−1′wr,r−2,w1…wr′,r and Nw1…wr′,r−1=w1…wr,r,w1…wr′,r−2,w1…wr−1′wr′,r−2,w1…wr′,r.
Thus,τv=9 for v∈Vr. On the other hand for v∈Vr, dv=4. So, ∑v∈Vrdvτv=4.9.2r=36.2r.Case 3.
Fori=1,i=2r−1
Letv∈V1, that is v=w1w2…wr,1. Then Nv=w1…wr,2,w1w2′…wr,2,w1…wr,0,w1′…wr,0. Furthermore, Nw1…wr,2=w1…wr,1,w1…wr,3,w1w2′…wr,1,w1w2w3′…wr,3, Nw1w2′…wr,2=w1…wr,1,w1w2′…wr,3,w1w2′…wr,1,w1w2′w3′…wr,3, Nw1w2…wr,0=w1w2…wr,1,w1′w2…wr,1 and.
Nw1′…wr,0=w1w2…wr,1,w1′w2…wr,1. Thus, τv=6 for v∈V1. On the other hand for v∈V1, dv=4. So, ∑v∈V1dvτv=4.6.2r=24.2r. The case for i=2r−1 is similar.Case 4.
Fori=r−1,i=r+1
Letv∈Vr−1, that is v=w1…wr,r−1. Then Nv=w1…wr,r,w1…wr′,r,w1…wr,r−2,w1…wr−1′wr,r−2. Furthermore, Nw1…wr,r=w1…wr,r−1,w1…wr,r+1,w1…wr′,r+1,w1…wr′,r−1, Nw1…wr′,r=w1…wr,r−1,w1…wr′,r+1,w1…wr′,r−1,w1…wr,r+1, Nw1…wr,r−2=w1…wr,r−1,w1…wr−2′wr−1wr,r−3,w1…wr,r−3,w1…wr−1′wr,r−1 and Nw1…wr−1′wr,r−2=w1…wr,r−1,w1…wr−1′wr,r−1,w1…wr−2′wr−1′wr,r−3,w1…wr−1′wr,r−3.
Thusτv=8 for v∈Vr−1. On the other hand for v∈Vr−1, dv=4. So ∑v∈Vr−1dvτv=4.8.2r=32.2r. The case for i=r+1 is similar.Case 5.
For1<i<r−1,r+1<i<2r−1
Letv∈Vi, where 1<i<r−1 or r+1<i<2r−1, that is v=w1…wr,i. Then Nv=w1…wr,i+1,w1…wi+1′…wr,i+1,w1…wr,i−1,w1…wi′…wr,i−1. Furthermore, Nw1…wr,i+1=w1…wr,i,w1…wr,i+2,w1…wi+2′…wr,i+2,w1…wi+1′…wr,i, Nw1…wr,i−1=w1…wr,i,w1…wr,i−2,w1…wi−1′…wr,i−2,w1…wi′…wr,i, Nw1…wi+1′…wr,i+1=w1…wr,i,w1…wi+1′…wr,i+2,w1…wi+1′…wr,i,w1…wi+1′wi+2′…wr,i+2 and Nw1…wi′…wr,i−1=w1…wr,i,w1…wi′…wr,i−2,w1…wi′…wr,i,w1…wi−1′wi′…wr,i−2. Thus τv=10, on the other hand dv=4 for v∈Vi. So ∑v∈Vidvτv=4.10.2rr−3=40.2rr−3. Hence ZC1∗G=∑v∈VGdvτv=2.10.2r+36.2r+2.32.2r+2.24.2r+2.40r−32r=2r+242+20r−3.
Also,ZC1G=2.25.2r+81.2r+2.36.2r+2.64.2r+2.100.2rr−3=200r.2r−269.2r.Theorem 2.
Forr−dimensional Butterfly networkG, we have(1)
ZC1∗G=2r40r−52(2)
ZC1G=100r.2r−178.2rPROOF.
Letv∈Vr in a butterfly network G, that is v=w1…wr,r. Then Nv=w1…wr,r−1,w1…wr′,r−1. Furthermore, Nw1…wr,r−1=w1…wr′,r,w1…wr,r,w1…wr,r−2,w1…wr−1′wr,r−2 and
Nw1…wr′,r−1=w1…wr−1′wr′,r−2],[w1…wr′,r][[w1…wr′,r−2],[w1…wr,r. Thus, for v∈Vr, τv=5 and dv=2. The case when v∈V0 is also same.
Moreover, ifv∈Vr−1 that is v=w1w2…wr,r−1. Then Nv=w1…wr,r,w1…wr−1′wr,r−2,w1…wr,r−2,w1…wr′,r and Nw1…wr−1′wr,r−2=w1…wr,r−1,w1…wr−1′wr,r−1,w1…wr−1′wr,r−3w1…wr−2′wr−1′wr,r−3Nw1…wr,r−2=w1…wr,r−1,w1…wr−1′wr,r−1,w1…wr−2′wr−1wr,r−3,w1…wr,r−3, Nw1w2…wr,r=w1w2…r,r−1,w1…wr′,r−1 and. Nw1…wr′,r=w1w2…wr,r−1,w1…wr′,r−1.
Thus, forv∈Vr−1, τv=6 and dv=4. The case for v∈V1 is also similar.
By following the same pattern, it can be proved that ifv∈Vi for 2≤i≤r−2, dv=4 and τv=10. So, ZC1∗G=2.5.2r+4.6.2r+4.10.2rr−3+2.5.2r+4.6.2r=2r40r−52 and ZC1G=25.2r+36.2r+100.2rr−3+36.2r+25.2r=100r.2r−178.2r.


## 2.2. Modified Zagreb Connection Indices for Vertical Cylindrical Representations of Benes Networks

This network is obtained by the identification of last column with first column of Benes network. Following the construction ofVCBr, clearly there are r2r+2 vertices in VCBr. This is a regular graph with degree of each vertex as four [9].Theorem 3.
ForG=VCBr, we have(1)
ZC1∗G=2r200+80r−3(2)
ZC1G=200r.2r−182.2rPROOF.
LetG denotes the 4-regular graph of VCBr in which w,i=w1w2…wr,i is an arbitrary node, for any 0≤i≤2r−1. Let Vk, 0≤k≤2r−1 be the set of vertices of k th column in G. For 0≤i,j≤2r−1, if w,i and w′,j are adjacent then wj=0 implies wj′=1, and vice versa. We partition the vertices v=w,i, 0≤i≤2r−1 of G in to the different cases depending on τv:Case 6.
Fori=0
Letv∈V0, that is v=w1w2…wr,0, then Nv=Nw1…wr,0=w1…wr,1,w1′…wr,1,w1…wr,2r−1,w1′…wr,2r−1.
Furthermore,Nw1w2…wr,1=w1…wr,0,w1…wr,2,w1′…wr,0,w1w2′…wr,2Nw1′…wr,1=w1…wr,0,w1′…wr,2,w1′…wr,0,w1′w2′…wr,2, Nw1…wr,2r−1=w1…wr,2r−2,w1w2′…wr,2r−2,w1′…wr,0,w1…wr,0 and. Nw1′…wr,2r−1=w1′…wr,2r−2,w1′…wr,0,w1′w2′…wr,2r−1,w1…wr,0. So, τv=9 and therefore, ∑v∈V0dvτv=4.9.2r=36.2r.Case 7.
Fori=1,i=2r−1
Letv∈V1, that is v=w1w2…wr,1. Then Nv=w1…wr,2,w1w2′…wr,2,w1…wr,0,w1′…wr,0.
Furthermore,Nw1w2…wr,2=w1…wr,1,w1…wr,3,w1w2w3′…wr,3,w1w2′…wr,1, Nw1w2′…wr,2=w1…wr,1,w1w2′…wr,3,w1w2′…wr,1,w1w2′w3′…wr,3,
Nw1…wr,0=w1…wr,2r−1],[w1′…wr,2r−1][[w1′…wr,1],[w1…wr,1 and Nw1′…wr,0=w1′…wr,2r−1,w1…wr,2r−1,w1′w2′…wr,2r−1,w1…wr,1. So, τv=8 and therefore ∑v∈V1dvτv=4.8.2r=32.2r. The case for i=2r−1 is similar.Case 8.
For1<i<2r−1
Similarly, ifv∈Vi, where 1<i<2r−1, then we have ∑v∈Vi=2.40r−32r+2.32.2r+36.2r=2r80r−3+100. Therefore, ZC1∗G=∑v∈VGdvτv=36.2r+2.32.2r+2r80r−3+100=2r200+80r−3 and ZC1G=81.2r+2.64.2r+2.100r−3.2r+2.64.2r+81.2r=200r.2r−182.2r.


## 2.3. Modified Zagreb Connection Indices for Horizontal Cylindrical Representations of Benes Networks

The networkHCBr is obtained by identifying vertices of the last row of Benes network with the corresponding vertices of the first row. The total vertices in HCBr are 2r+12r−1 [9]. By following the same method as in previous theorems, the 2r+12r−1 vertices are partitioned in terms of degree and τ as shown in Table 1.Table 1
Vertex partition ofHCBr corresponding to connection numbers and degrees.
DegτvVertices3926112614261616182 (r − 3)252 (2r − 34)272 (2)462 (2r − 36)482 (2)472 (2)4822r−64102 (4)4122492r−441022r−4 (r − 33)4142 (2)Theorem 4.
ForG=HCBr, we have(1)
ZC1∗G=802+104r−72.2r+80r.2r(2)
ZC1G=200r.2r−269.2r−152r+2304PROOF.
From the partition of the vertices ofHCBr and Table 1 comprising connection number of the vertices, we have
ZC1∗G=∑v∈VGdvτv=54+132+168+96+216r−3+202r−4+56+482r−6+128+112+642r−6+320+96+362r−4+802r−4r−3+224=1386+216r−648+20.2r−80+48.2r−288+64.2r−384+36.2r−144+80r.2r−r320−240.2r+960=802−104r−72.2r+80r.2r.
Similarly,ZC1G=162+242+392+256+648r−1944+50.2r−200+196+72.2r−432+256+196+128.2r−768+800+288+81.2r−324+200.r.2r−800r−600.2r+2400+784=200r.2r−269.2r−152r+2304.


## 2.4. Modified Zagreb Connection Indices for Toroidal Representation of Benes Networks

TheTBr is obtained by the identification of the vertices of bottom row of VCBr to the vertices of the top row. Here, benes network is embedded on Torus. The total number of vertices in TBr is 2r2r−1 [9]. Now, we compute ZC1∗ and ZC1 for this network.Theorem 5.
ForG=TBr, we have(1)
ZC1∗G=80r.2r−40.2r−104r+1044(2)
ZC1G=200r.2r−182.2r−152r+3020PROOF.
The2r2r−1 vertices of TBr are partitioned on the basis of degree as: 2r2r−2 and 2r vertices have degrees 4 and 6, respectively. The connection numbers of vertices of TBr are presented in Table 2.
So, we have
ZC1∗G=∑v∈VGdvτv=108+168+168+96+216r−3+362r−4+192+642r−4+176+112+642r−6+320+96+362r−4+802r−4r−3+224=80r.2r−40.2r−104r+1044.
Moreover,ZC1G=324+392+392+256+648r−1944+81.2r−324+576+128.2r−512+484+196+128.2r−768+800+288+81.2r−324+200.r.2r−800r−600.2r+2400+784=200r.2r−182.2r−152r+3020.Table 2
Vertex partition ofTBr corresponding to connection numbers and degree.
DegτvNo. of vertices61816142614261616182 (r-3)49(2r-4)4122 (2)482 (2r-4)4112 (2)472 (2)4822r−64102 (4)4122492r−441022r−4 (r-3)4142 (2)


## 3. Conclusion

The newly generated structures and networks are always interesting topic to be studied. In [9], several new networks such as HCBr, VCBr and TBr have been introduced by using Br. 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 Br, BFr, VCBr, HCBr, and TBr. The current paper will be a step forward towards the study of these networks for general distance and the maximum distance based descriptors.Figure 7
Comparison graph.Figure 8
Comparison graph.



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