Calculating topological indices of certain OTIS interconnection networks

Adnan Aslam; Safyan Ahmad; Muhammad Ahsan Binyamin; Wei Gao

doi:10.1515/chem-2019-0029

Article Open Access

Calculating topological indices of certain OTIS interconnection networks

Adnan Aslam , Safyan Ahmad , Muhammad Ahsan Binyamin and Wei Gao

Published/Copyright: April 10, 2019

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From the journal Open Chemistry Volume 17 Issue 1

Abstract

Recently, increasing attention has been paid to The Optical Transpose Interconnection System (OTIS) network because of its prospective applications in architectures for parallel as well as distributed systems [27, 28]. Different interconnection networks in the context of topological indices are researched recently in [25, 26]. This article includes the computions of the general Randi´c, first and second Zagreb, general sum connectivity, first and second multiple zagreb, hyper zagreb, ABC and GA indices for OTIS (swapped and biswapped) networks by taking path and k-regular graph on n vertices as a base graphs. In addition, some delicated formulas are also obtained for the ABC₄ and GA₅ indices for the OTIS biswapped networks by considering basis graph as a path and k-regular graph of order n.

Keywords: Optical Transpose Interconnection System (OTIS) network; OTIS swapped networks; OTIS biswapped networks; molecular topological indices

1 Introduction and Preliminaries

As a new emerging science, Cheminformatics is related to chemistry, mathematics and computer sciences, whose major components include Quantitative structure-activity (QSAR) and structure-property relationships (QSPR) and the components can contribute to the research on physicochemical properties of chemical compounds.

As a numeric quantity, a topological index is closely related to a graph which is invariant under graph automorphism and can characterize the topology of a graph. Numerous applications of graph theory can be found in structural chemistry. Its first well-known application in chemistry was the study of paraffin boiling points by Wiener([35]). Various topological indices were introduced following this study that explained physico-chemical properties.

Originally, the main function of optical transpose interconnection system (OTIS) networks was to offer efficient connectivity for new optoelectronic computer architectures which benefit from both optical and electronic technologies [24]. In OTIS networks, processors are organized into clusters. Electronic interconnects are utilised between processors within the same cluster, while optical links are utilised for intercluster communication. Numerous algorithms have been devised for routing, selection/sorting [13, 16, 30, 31], certain numerical computations [29], Fourier transform [7], matrix multiplication [15], and image processing [14].

The structure of an interconnection network can be mathematically modeled by a graph. The vertices of this graph represent the processor nodes and the edges represent the links between the processors. The topology of a graph determines the way in which vertices are connected by edges. From the topology of a network, certain properties can easily be determined. The diameter is determined as the maximum distance between any two nodes in the network. The number of links connected to a node determines the degree of that node. If this number is the same for all nodes in the network, the network is called regular.

In this article, G is considered to be a simple graph with the vertex set V(G) and the edge set E(G). Since u ∈ V(G), we denote N_u as the set of its neighbors in G, the degree d_u of vertex u is d_u = |N_u| and Su=∑v∈Nudv.The first degree based topological index is Randić index brought about by Milan Randić [32] in 1975. The general Randić connectivity index [9] of G is denoted as

(1)Rα(G)=∑uv∈E(G) (dudv)α,

where α is a real number. When α=−12,then R_−1/2(G) is known as Randić connectivity index of G.

The sum-connectivity index was introduced by Bo Zhou and Nenad Trinajstić [10] to be described as

(2)SCI(G)=∑uv∈E(G)1du+dv.

The general sum-connectivity index χ_α(G) was proposed in 2010 in [11]:

(3)χα(G)=∑uv∈E(G)(du+dv)α.

Approximate four decades ago, Ivan Gutman and Trinajstić proposed an significant topological index. To be simple, first Zagreb index denoted by M₁(G) was denoted as the sum of degrees of end vertices of all edges of G. We describe the first Zagerb M₁(G) and second Zagreb M₂(G) is defined as

(4)M1(G)=∑v∈V(G)dv2=∑uv∈E(G)[du+dv]

(5)M2(G)=∑uv∈E(G)[du×dv]

where d(u) is the degree of the vertex u in the graph G.

Ghorbani et al [21] described two new versions of the Zagreb indices of a graph G in 2012 to be the first multiple Zagreb index PM₁(G), second multiple Zagreb index PM₂(G), which are denoted as

(6)PM1(G)=∏uv∈E(G)[du+dv]

(7)PM2(G)=∏uv∈E(G)[du×dv]

Currently, Shirdel et al [33] raised up the hyper-Zagreb index:

(8)HM(G)=∑uv∈E(G)[du+dv]2

Atom-bond connectivity (ABC) index proposed by Estrada et al in [19] have numerous applications for its connectivity topological index. The ABC index of graph G is defined as

(9)ABC(G)=∑uv∈E(G)du+dv−2dudv.

The fourth member of the class of ABC index is ABC₄ was introduced by M. Ghorbani et al in [22] as:

(10)ABC4(G)=∑uv∈E(G)Su+Sv−2SuSv.

D. Vukicevic and B. Furtula raised up the geometric arithmetic (GA) index in [34] and described as

(11)GA(G)=∑uv∈E(G)2dudvdu+dv.

Currently, the fifth version of GA index is (GA₅) is put forward by Graovac et al in [23] as

(12)GA5(G)=∑uv∈E(G)2SuSvSu+Sv.

In 2014, Hayat et al computed the detailed expressions for the ABC, GA, ABC₄ and GA₅ indices for diverse networks like silicates, honeycomb, hexagonal and oxides in [25]. They also studied the general Randić, ABC, GA and first Zagreb indices for butterfly and benes networks in [26]. For detailed results on the topological indices of graphs the readers may refer to [1, 2, 3, 4, 5, 6, 8, 12, 17, 18, 37, 38].

The following presents an example of how the indices can be used in chemical engineering.

Example The generalized first Zagerb index is formulated as ∑v∈V(G)dvαwhere α is a real number. When α = 2 it becomes an equation (4) and when α = 3 it becomes a forgotten index which is stated as

F(G)=∑v∈V(G)dv3=∑uv∈E(G)(du2+dv2).

As we can see in website http://www.moleculardescriptors.eu/dataset/dataset.htm in accordance to the International Academy of Mathematical Chemistry, the potential ability of this index was tested in terms of a data set of octane isomers. In the simplest form, the F index doesn’t identify multiple bonds and hetero atoms, which becomes the reason why a data set is selected as a measure. A list of data including entropy, heat of vaporization, boiling point, density, heat capacities, melting point, enthalpy of formation, motor octane number, acentric factor, octanol-water partition coefficient, molar refraction, total surface area, and molar volume help to compose the octane data set. The F index indicates its strong bonds with most characteristics .As a result, the F index is verified to have correlation coefficients larger than 0.95 in the entropy and acentric factor.

For various other chemical features, whereas, F index may not be strongly related. To strengthen the predictive ability of the F index in possible chemical application circumstances, a linear framework was introduced as follows (see Furtula and Gutman [20]):

∑uv∈E(G)(du+dv)+λ∑uv∈E(G)(du2+dv2),

where λ is a balanced parameter adjusted to the specific applications in chemical engineering (generally speaking, λ always takes a value in interval [−20, 20]); the first term Σ_uv∈_E(_G₎(d_u + d_v) is the first Zagreb index described in (4). In light of massive experimental studies, this framework can be applied in every chemical property with a certain octane database. For instance, a significant improvement can be yielded in the octanol-water partition coefficient, and it’s revealed that the absolute value of the correlation coefficient infers a sharp maximum by taking λ = −0.14 in the above computing model. log P is the logarithm function of the octanol-water partition coefficient. Hence, by means of derivation, the octanol-water partition coefficient of octanes can be formulated as follows:

logP=−0.2058(∑uv∈E(G)(du+dv) −0.14∑uv∈E(G)(du2+dv2)) + 7.5864.

This fact indicates that the correlation coefficient can reach 0.99896 and the mean absolute percentage error is only 0.06%.

The paper is structured below: In Section 2, we compute the Randić, first and second Zagreb , hyper Zagreb, first and second multiple Zagreb, general sum connectivity, ABC, ABC₄, GA and GA₅ indices for OTIS swapped networks by taking path and k-regular graphs on n vertices as a base graph. In Section 3, we give explicit formulas of these indices for the OTIS biswapped networks by considering a basis graph as a path and k-regular graph of order n.

2 Topological indices of OTIS swapped networks

Definition 2.1The OTIS swapped network O_Ω, deduced from the graph Ω, is a graph with vertex set V(O_Ω) = {〈g, p〉|g, p ∈ V(Ω)} and edge set E(O_Ω) = {(g, p₁〉, 〈g, p₂〉)|g ∈ V(Ω), (p₁, p₂) ∈ E(Ω)} ∪ {(g, p〉, 〈p, g〉)|g, p ∈ V(Ω) and g ≠ p}

In OTIS swapped network O_Ω, the graph Ω is named the basis (factor) graph or network. If the basis network Ω has n nodes, then O_Ω is composed of n node-disjoint subnetworks called clusters, which are isomorphic to Ω. The node label 〈g, p〉 in O_Ω define the node indexed p in cluster g. The vertex and edge set cardinalities of OTIS Swapped network O_Ω are n² and n|E(Ω)|+n(n−1)2respectively, where n is the number of vertices in Ω. Figure 1 depicts an OTIS swapped network with a complete graph of order 6 (K₆) as the basis graph.

Figure 1

OTIS swapped network O_K6

2.1 Results for OTIS Swapped networks OP_n

Set P_n as a path on n vertices and O_Pn as OTIS swapped network with basis network P_n. An OTIS swapped network with the basis network P₆ is shown in Figure 2.

Now the certain degree based topological indices of OTIS swapped network O_Pn is computed.

Theorem 2.2

The general Randić index and the general sum connectivity index of O_Pn is equal to

Rα(OPn)=2×3α+3×4α+(6n−14)×6α +3(n−2)(n−3)2×9α,

χα(OPn)=5×4α+(6n−14)×5α+3(n−2)(n−3)2×6α.

Proof The number of vertices and edges in O_Pn are n² and 3n(n−1)2respectively. There are four types of edges in

Figure 2

OTIS swapped network O_P6

O_Pn based on degrees of end vertices of each edge. Table 1 shows such an edge partition for O_Pn . Using formula (1) and (3) in Table 1, we obtain the expression of the indices.

Table 1

The edge partition of the graph O_Pn

(d_u , d_v) where uv ∈ E(G)	(1,3)	(2,2)	(2,3)	(3,3)
Number of edges	2	3	6n − 14	3(n−2)(n−3)2

From the above theorem, one can immediately compute Randić connectivity, first and second Zagreb, hyper Zagreb and sum connectivity indices of OTIS swapped network O_Pn .

Corollary 2.3Let G be the graph of OTIS swapped network O_Pn . Then

R−12(G)=n22+(6−52)n+23−146+92, M1(G)=9n2−15n+4, M2(G)=27n2−63n+302, HM(G)=54n2−120n+106,SCI(G)=3n2−15n+18+5626+6n−145.

In the theorem below, we calculate first and second multiple Zagreb, atom bond connectivity and geometric arithmetic indices of the OTIS swapped network O_Pn .

Theorem 2.4

Consider an OTIS swapped network O_Pn, then

PM1(OPn)=8640(3n2−22n+53n−42),

PM2(OPn)=11664(3n3−22n+53n−42),ABC(OPn)=n2+n(18−5)+223+322+6−1436, GA(OPn)=3n22+n(1265−152)+9+3−2865.

Proof The edge partition based on the degree of end vertices of each edge is shown in the Table 1. We apply formulas (6), (7), (9) and (11) to the information in Table 1 and obtain the required results.

2.2 Results for OTIS (Swapped) networks OR_k

Let R _k be k-regular graph on n vertices and O _Rk be the OTIS swapped network with the basis network R_k. Figure 1 depicts an example of OTIS swapped network O_K6. Now we calculate certain degree based topological indices of the OTIS swapped network O_Rk .

Theorem 2.5

Let O_Rk be an OTIS swapped network, then its general Randić and the general sum connectivity index is equal to

Rα(ORk)=nk(k2+k)α+n2(k+1)−n(2k+1)2(k+1)2α,χα(ORk)=nk(2k+1)α+n2(k+1)−n(2k+1)2(2k+2)α.

Proof The number of vertices and edges in O_Rk are n² and n2(k+1)−n2respectively. Two sorts of edges exist in O_Rk which are based on degrees of the end vertices of each edge. Such an edge partition of O_Rk can be found in Table 2. Using formula (1) and (3) in Table 1, we obtain the expression of the indices.

Table 2

The edge partition of the graph O_Rk

(d_u , d_v) where uv ∈ E(G)	(k, k + 1)	(k + 1, k + 1)
Number of edges	nk	n2(k+1)−n(1+2k)2

From the above theorem, one can compute Randić connectivity, first and second Zagreb, hyper Zagreb and sum connectivity indices of OTIS swapped network O_Rk .

Corollary 2.6Let G be the graph of OTIS swapped network O_Rk . Then

R−12(G)=nkk2+k+n2(k+1)−n(2k+1)2k+2,

M1(G)=n2(k+1)2−n(2k+1),M2(G)=n2(k+1)3−n(k+1)(3k−1)2,HM(G)=2n2(k+1)3−n(2k+1)(3k+2),SCI(G)=nk2k+1+n2(k+1)−n(2k+1)22k+2.

In the theorem below, we calculate first and second multiple Zagreb, atom bond connectivity and geometric arithmetic indices of the OTIS swapped network O_Rk .

Theorem 2.7

Consider an OTIS swapped network O_Rk , then

PM1(ORk)=nk(k+1)(2k+1)(n2(k+1)−n(2k+1)), PM2(ORk)=nk2(k+1)3(n2(k+1)−n(2k+1))2,ABC(ORk)=2kn22+n(k2k−1k2+k−2k(2k+1)2(k+1)), GA(ORk)=n2(k+1)2+n(2kk2+k2k+1−2k+12).

Proof The edge partition based on the degree of end vertices of each edge is described in the Table 2. We practice formulas (6), (7), (9) and (11) to the information in Table 2 and obtain the expected results.

3 Topological indices of Biswapped networks

For a base graph Ω, the biswapped interconnection network Bsw(Ω) is a graph with vertex set and edge set specified as:

V(Bsw(Ω))={〈0,p,g 〉,〈1,p,g 〉|p,g∈V(Ω)}E(Bsw(Ω))={(〈0,p,g1 〉,〈0,p,g2 〉),(〈1,p,g1 〉, 〈1,p,g2 〉)|(g1,g2)∈E(Ω),p∈V(Ω)} ∪{(〈0,p,g 〉,〈1,g,p 〉)|p,g∈V(Ω)}

The definition postulate 2n clusters, whereby each cluster is a Ω graph. The n clusters with nodes labelled 0, cluster^♯, node^♯〉 form part 0 of the bipartite graph and the remaining n cluster constitutes part 1 with associated nodes numbers 1, cluster^♯, node^♯〉. Each cluster p in either part of Bsw(Ω) has the similar internal connectivity as Ω. Moreover, node g of cluster p in part 0 is connected to node p of cluster g of part 1.

As an example, when Ω = C₄ constitutes the basis graph then Figure 3 represent the resulting Bsw(Ω). Part 0 of the network is drawn at the top and part 1 at the bottom, with cluster 0 − 3 positioned from left to right. The vertex and edge set cardinalities of biswapped network Bsw(Ω) are 2n² and 2n|E(Ω)|+n² respectively, where n is the number of vertices in Ω. Some topological properties of the biswapped network are studied in [36].

Now we calculate certain degree based topological indices of the biswapped network.

3.1 Results for Biswapped networks Bsw(Pn)

Let P_n be path on the n vertices and Bsw(P_n) be the biswapped network with the basis network P_n. The number of vertices and edges in Bsw(P_n) are 2n² and 3n² −2n respectively. Figure 4 shows a biswapped network with the 5-node path P₅ as the basis graphs. We compute the general Randić and the general sum connectivity index in the theorem below.

Theorem 3.1

Let Bsw(P_n) be a biswapped network, then its general Randić and general sum connectivity index equals to

Rα(Bsw(Pn))=4×4α+8(n−1)6α+(3n2−10n+4)9α,χα(Bsw(Pn))=4×4α+8(n−1)5α+(3n2−10n+4)6α.

Proof The number of vertices and edges in Bsw(P_n) are 2n² and 3n² − 2n respectively. There are three types of edges in Bsw(P_n) based on degrees of end vertices of each edge. Table 3 reveals such an edge partition of Bsw(P_n). Using formulas (1) and (3) in table 3 , we obtain the expression of the indices. From the above theorem, one can

Table 3

The edge partition of the graph Bsw(P_n)

(d_u , d_v) where	(2,2)	(2,3)	(3,3)
uv ∈ E(Bsw(P_n))
Number of edges	4	8(n − 1)	3n² − 10n + 4

Figure 3

Biswapped network Bsw(C₄)

Figure 4

Biswapped network Bsw(P₅)

compute Randić connectivity, first and second Zagreb, hyper Zagreb and sum connectivity indices of OTIS swapped network Bsw(P_n).

Corollary 3.2Let G be the graph of OTIS swapped network Bsw(P_n). Then

R−12(G)=n2+(436−103)n+103−436, M1(G)=18n2−20n, M2(G)=27n2−42n+4, HM(G)=108n2−160n+8,SCI(G)=3n26+n(85−106)+2+46−15.

In the theorem below, we calculate atom bond connectivity, first and second multiple Zagreb, and geometric arithmetic indices of the OTIS swapped network Bsw(P_n).

Theorem 3.3

Consider an OTIS swapped network Bsw(P_n), then

PM1(Bsw(Pn))=3840(3n3−13n2+14n−4),PM2(Bsw(Pn))=6912(3n3−13n2+14n−4),

ABC(Bsw(Pn))=2n2+(42−203)n−22+83,GA(Bsw(Pn))=3n2+(1656−10)n+8−1656.

Proof The edge partition based on the degree of end vertices of each edge is described in the Table 3. We practice formulas (6), (7), (9) and (11) to the information in Table 3 and obtain the expected results.

Theorem 3.4

Consider n ≥ 5 and Bsw(P_n) a biswapped network with basis network P_n, then

ABC4(Bsw(Pn)) =43n2+(−539+46221+3414+1330)n +852−13714+409+27182−2330.

GA5(Bsw(Pn)) =3n2+(41342+2473−14+48172)n +8+4335−6473+321514−96172.

Proof Let Bsw(P_n) be a biswapped network with basis network P_n. We prove it by considering the edge partition in Table 4. We practice formulas (10) and (12) to the information provided in Table 4 and obtain the expected results.

Table 4

The edge partition of the graph Bsw(P_n)

(S_u , S_v) where uv ∈ E(G)	Number of edges	(S_u , S_v) where uv ∈ E(G)	Number of edges
(5,5)	4	(5,7)	8
(6,7)	2n	(6,8)	6n − 16
(7,8)	8	(8,8)	6n − 24
(8,9)	4n − 8	(9,9)	3n² − 20n + 28

3.2 Results for Biswapped networks Bsw(Rk)

Let R_k be k-regular graph of order n and let Bsw(R_k) be the biswapped network with the basis network R _k. We can see that Bsw(R_k) is a k + 1 regular graph of order 2n and size n²(k+1). A biswapped network with the basis network C⁴ is shown in Figure 3. We compute the general Randić and the general sum connectivity index in the following theorem.

Theorem 3.5

Let Bsw(R_k) be a biswapped network, then its general Randić and general sum connectivity index is equal to

Rα(Bsw(Rk))=n2(k+1)2α+1,χα(Bsw(Rk))=2αn2(k+1)α+1.

Proof The number of vertices and edges in Bsw(R_k) are 2n and n²(k + 1) respectively. In the biswapped network Bsw(R_k), deg(u) = k + 1 for every vertex u in Bsw(R_k). Using formulas (1) and (3), we obtain the expression of the indices. The Randić connectivity, first and second Zagreb, first and second multiple Zagreb, hyper Zagreb and sum connectivity indices of OTIS swapped network Bsw(R_k) is sated in the following corollary.

Corollary 3.6Let G be the graph of the OTIS swapped network Bsw(R_k). Then

R−12G=n2,PM1G=M1G=2n2k+12,PM2G=M2G=n2k+13,HMG=4n2k+13,

SCI(G)=n2(k+1)2k+2.

Next, the atom-bond connectivity index and fourth version of atom-bond connectivity index of Biswapped network Bsw(R_k) is caculated.

Theorem 3.7

Let Bsw(R_k) be a biswapped network, then

ABC(Bsw(Rk))=n22k,ABC4(Bsw(Rk))=n22k2+4kk+1.

Proof The biswapped network is a k +1 regular graph i.e., the set of neighbors is the same for every vertex. The sum of degrees of all vertices adjacent to every vertex u is (k + 1)². Thus du+dv−2dudv=2kk+1 and Su+Sv−2SuSv=2k2+4k(k+1)2for every edge uv ∈ Bsw(R_k). Then according to formula (9), we have

ABC(Bsw(Rk)) =∑uv∈E(Bsw(Rk))du+dv−2dudv =n2(k+1).2kk+1=n22k

and according to formula (10) we obtain

ABC4(Bsw(Rk)) =∑uv∈E(Bsw(Rk))Su+Sv−2SuSv =n2(k+1)⋅2k2+4k(k+1)2=n22k2+4kk+1.

Hence, we finish the proof. In the theorem below, we determine the geometric-arithmetic GA index and fifth version of the GA index of the biswapped network.

Theorem 3.8

Consider a Biswapped network Bsw(R_k), then

GA(Bsw(Rk))=n2(k+1),GA5(Bsw(Rk))=n2(k+1)2.

Proof It is easy to see that 2dudvdu+dv=1 and 2SuSvSu+Sv=k+1for every edge uv ∈ Bsw(R_k). Then using formula (11) for the geometric-arithmetic index we have

GA=(Bsw(Rk))=∑uv∈E(Bsw(Rk))2dudvdu+dv=n2(k+1).

According to Formula (12) for the fifth version of geometric-arithmetic index we obtain

GA5(Bsw(Rk))=∑uv∈E(Bsw(Rk))2SuSvSu+Sv=n2(k+1)2.

The desired result is proved.

4 Conclusion

This paper addressed the OTIS swapped networks and biswapped networks and researched on their topological indices. We determined the general Randić, general sum connectivity, first and second Zagreb, first and second multiple Zagreb, hyper Zagreb atombond and geometric indices for both the family of networks by considering the basis network as path P_n and k-regular graph R _k. We also gave explicit formulae for ABC₄ and GA₅ indices of these networks with the basis network R _k.

Acknowledgement

This work has been partially supported by National Science Foundation of China (no. 11761083).

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Received: 2018-04-21

Accepted: 2018-10-10

Published Online: 2019-04-10

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/chem-2019-0029

Keywords for this article

Optical Transpose Interconnection System (OTIS) network; OTIS swapped networks; OTIS biswapped networks; molecular topological indices

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BY 4.0

Calculating topological indices of certain OTIS interconnection networks

Article

Abstract

1 Introduction and Preliminaries

2 Topological indices of OTIS swapped networks

2.1 Results for OTIS Swapped networks OPn

Theorem 2.2

Theorem 2.4

2.2 Results for OTIS (Swapped) networks ORk

Theorem 2.5

Theorem 2.7

3 Topological indices of Biswapped networks

3.1 Results for Biswapped networks Bsw(Pn)

Theorem 3.1

Theorem 3.3

Theorem 3.4

3.2 Results for Biswapped networks Bsw(Rk)

Theorem 3.5

Theorem 3.7

Theorem 3.8

4 Conclusion

Acknowledgement

References

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2.1 Results for OTIS Swapped networks OP_n

2.2 Results for OTIS (Swapped) networks OR_k