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Topology-Based Analysis of OTIS (Swapped) Networks [mathematical expression not reproducible].

1. Introduction

Cheminformatics is a new branch of science which relates chemistry, mathematics, and computer sciences. Quantitative structure-activity relationship (QSAR) and quantitative structure-property relationship (QSPR) are the main components of cheminformatics which are helpful to study physicochemical properties of chemical compounds [1-3].

A topological index is a numeric quantity associated with the graph of chemical compound, which characterizes its topology and is invariant under graph automorphism [4-6]. There are numerous applications of graph theory in the field of structural chemistry. The first well-known use of a topological index in chemistry was by Wiener in the study of paraffin boiling points [7-9]. After that, in order to explain physicochemical properties, various topological indices have been introduced and studied [10, 11].

A computer network is a digital telecommunications network which allows nodes to share resources. In computer networks, computing devices exchange data with each other using connections (data links) between nodes. These data links are established over cable media such as wires or optic cables, or wireless media such as WiFi. Optical transpose interconnection system (OTIS) networks were initially contrived to give productive network to new optoelectronic computer models that profit by both optical and electronic advancements [12]. In OTIS networks, processors are orchestrated into groups. Electronic inter-connects are used between processors within the same cluster, while optical links are used for intercluster communication. Various algorithms have been produced for directing, determination/arranging, certain numerical calculations, Fourier transformation [13], matrix multiplication [14], image processing [15], and so on [16, 17]. The structure of an interconnection system can be scientifically modeled by a graph. The vertices of this graph are the processor nodes and the edges are the connections between the processors. The topology of a graph decides the manner by which vertices are associated by edges. From the topology of a system, certain properties can be decided. The diameter of a graph is the maximum distance between any two vertices of the graph.

Definition 1 (OTIS (swapped) network [mathematical expression not reproducible]). The OTIS (swapped) network is derived from the graph [OMEGA], which is a graph with vertex set V([O.sub.[OMEGA]]) = < g,p > | g,p [member of] V([OMEGA]) and edge set E([O.sub.[OMEGA]]) = < g, [p.sub.1] >, < g, [p.sub.2] > g [member of] V([OMEGA]), ([p.sub.1], [p.sub.2]) [member of] E([OMEGA]) [union] (< g, p >, < g, p, g >)| g, p [member of] V([OMEGA]) and g [not equal to] p.

The graph of OTIS (swapped) network [mathematical expression not reproducible] given in Figure 1 has ([n.sup.3]/2) + 3[n.sup.2] + (11n/2) + 3 edges and [(n + 2).sup.2] vertices.

Definition 2 (OTIS (swapped) network [mathematical expression not reproducible]). Let [P.sub.n] be path of n vertices and O[P.sub.n] be OTIS (swapped) network with basis network [P.sub.n]. An OTIS (swapped) network with the basis network [P.sub.6] is shown in Figure 2.

The OTIS (swapped) network graph [mathematical expression not reproducible] given in Figure 2 has (3/2)([n.sup.2] - n) edges and [n.sup.2] vertices.

In this paper, we aim to compute degree-dependent topological indices of OTIS (swapped) networks [mathematical expression not reproducible]. In Section 2, we give definitions and literature review about TIs. In Sections 3 and 4, we present the methodology and application of our results in chemistry, respectively. Section 5 contains our main results, and Section 6 concludes our paper.

2. Topological Indices (TIs)

TIs are numbers which depend on the molecular graph and are helpful in deciding the properties of the concerned molecular compound [18-20]. We can consider TI as a function which assigns a real number to each molecular graph, and this real number is used as a descriptor of the concerned molecule. From the TIs, a variety of physical and chemical properties like heat of evaporation, heat of formation, boiling point, chromatographic retention, surface tension, and vapor pressure of understudy molecular compound can be identified. A TI gives us the mathematical language to study a molecular graph. There are three types of TIs:

(1) Degree-based TIs

(2) Distance-based TIs

(3) Spectrum-based TIs

The first type of TI depends upon the degree of vertices, second one depends upon the distance of vertices and the third type of TI depends upon the spectrum of graph.

2.1. Zagreb Indices. To compute total [pi]-electron energy, the following TI is defined:

[M.sub.1](G) = [summation over (v[member of]V(G))] [d.sup.2.sub.v]. (1)

But soon it was observed that this index increases with increases in branching of the skeleton of carbon atoms. After 10 years, Balaban et al. wrote a review [21], in which he declared M1 and M2 are among the degree-based TIs and named them as Zagreb group indices. The name Zagreb group indices was soon changed to Zagreb indices (ZIs), and nowadays, [M.sub.1] and [M.sub.2] are abbreviated as first Zagreb index and second Zagreb index.

In 1975, Gutman et al. gave a remarkable identity [22]. Hence, these two indices are among the oldest degree-based descriptors, and their properties are extensively investigated. The mathematical formulae of these indices are

[M.sub.1](G) = [summation over (uv[member of]E(G))] [d.sub.u] + [d.sub.v], [M.sub.2](G) = [summation over (uv[member of]E(G))] [d.sub.u][d.sub.v]. (2)

For detailed survey about these indices, we refer [23-26]. Doslic et al. [27] gave the idea of augmented ZI, whose mathematical formula is

AZI(G) = [summation over (uv[member of]E(G))] [([d.sub.u][d.sub.v]/[d.sub.u] + [d.sub.v] -2).sup.3]. (3)

2.2. Randic or Connectivity Index. Historically, ZIs are the very first degree-based TIs, but these indices were used for completely different purposes; therefore, the first genuine degree-based TI is the Randic index (RI) which was given in 1975 by Randic [28] as

[R.sub.-1/2](G) = [summation over (uv[member of]E(G))] 1/[square root of [d.sub.u][d.sub.v]]. (4)

Firstly, Randicc named it as a branching index, which was soon named as connectivity index, and nowadays, it is called as RI. The RI is the most popular degree-based TI and has been extensively studied by both mathematicians and chemists. Randicc himself wrote two reviews [29, 30], and many papers and books on this topological invariant are present in the literature; few of them are [31-34]. Researchers recognized the importance of the Randic index in drug design. Bollbas and Erdos, famous mathematicians of that time investigated some hidden mathematical properties of RI [35]; after that, RI was worth studying, and a surge of publications began [36-39]. An unexpected mathematical quality of the Randic index was discovered recently, which tells us about the relation of this topological invariant with the normalized Laplacian matrix [38, 40, 41]. The GRI known as general RI [42] is defined as

[R.sub.[alpha]](G)= [summation over (uv[member of]E(G))] [([d.sub.u][d.sub.v]).sup.[alpha]]. (5)

2.3. M-Polynomial. The mathematical formula of M-polynomial is

[mathematical expression not reproducible]. (6)

Detailed survey about the definitions of other TIs computed in this paper and relation of TIs with M-polynomial can be seen in [43, 44], where

[mathematical expression not reproducible]. (7)

3. Methodology

To compute the M-polynomial of a graph G, we need to compute the number of vertices and edges in it and divide the edge set into different classes with respect to the degrees of end vertices. From the M-polynomials, we can recover many degree-dependent indices by applying some differential and integral operators.

4. Applications in Chemistry

A topological description is used to depict the features of the studied compounds, and indices of graph-theoretical origin are used to investigate the correlations between structure and biological activity [45-48]. For example, the Randic index demonstrates great relationship with the physical property of alkanes. The geometric arithmetic index has a similar role as that of the Randicc index. The sum-connectivity index is helpful in guessing the melting point of compounds. Zagreb indices are used to calculate [pi]-electron energy.

5. Main Results

In this section, we compute M-polynomials of understudy networks and recover nine TIs from these polynomials.

5.1. Results for O[P.sub.n]

Theorem 1. Let O[P.sub.n] be the swapped network; then,

M (G; x, y) = 2x[y.sup.3] + 3[x.sub.2][y.sup.2] + (6n - 14)[x.sub.2][y.sup.3] + (3(n - 2)(n - 3)/2)[x.sup.3][y.sup.3]. (8)

Proof. The O[P.sub.n] network has the following four types of edges based on the degree of end vertices:

[mathematical expression not reproducible], (9)

such that

[mathematical expression not reproducible]. (10)

Now, from the definition of M-polynomial, we have

[mathematical expression not reproducible]. (11)

Corollary 1. Let O[P.sub.n] be the swapped network; then,

[M.sub.1](O[P.sub.n]) = 9[n.sup.2] - 15n + 4. (12)

Proof. Let

f (x, y) = M (G; x, y) = 2x[y.sup.3] + 3[x.sup.2][y.sup.2] + (6n - 14)[x.sup.2][y.sup.3] + (3(n - 2)(n - 3)/2)[x.sup.3][y.sup.3]. (13)

Then,

[mathematical expression not reproducible]. (14)

Now, we have

[M.sub.1](O[P.sub.n]) = [D.sub.x]f + [D.sub.y]f[|.sub.x=y=1] = 9[n.sup.2] - 15n + 4. (15)

Corollary 2. Let O[P.sub.n] be the swapped network; then,

[M.sub.2](O[P.sub.n]) = 27[n.sup.2]/2 63n/2 + 15. (16)

Proof. Let

f (x, y) = M (G; x, y) = 2x[y.sup.3] + 3[x.sup.2][y.sup.2] + (6n - 14)[x.sup.2][y.sup.3] + (3(n - 2)(n - 3)/2)[x.sup.3] [y.sup.3]. (17)

Then,

[mathematical expression not reproducible]. (18)

Now, we have

[M.sub.2](O[P.sub.n]) = [D.sub.x][D.sub.y]f|[x=y=1] = 27[n.sup.2]/2 63n/2 + 15. (19)

Corollary 3. Let O[P.sub.n] be the swapped network; then,

[sup.m][M.sub.2](O[P.sub.n]) = [n.sup.2]/6 + n/6 + 1/12. (20)

Proof. Let

f(x, y) = M (G; x, y) = 2x[y.sup.3] + 3[x.sup.2][y.sup.2] + (6n - 14)[x.sup.2][y.sup.3] + (3(n - 2)(n - 3)/2)[x.sup.3][y.sup.3]. (21)

Then,

[mathematical expression not reproducible]. (22)

Now, we have

[sup.m][M.sub.2](O[P.sub.n]) = [S.sub.x][S.sub.y] f[|.sub.x=y=1] = [n.sup.2]/6 + n/6 + 1/12. (23)

Corollary 4. Let OPn be the swapped network; then,

[R.sub.[alpha]](O[P.sub.n]) = [6.sup.[alpha]] + [6.sup.[alpha]][2.sup.[alpha]] + [3.sup.[alpha]][2.sup.[alpha]] (6n - 14) + [9.sup.[alpha]][3.sup.[alpha]] (n - 2)(n - 3)/[2.sup.[alpha]]). (24)

Proof. Let

f(x, y) = M (G; x, y) = 2x[y.sup.3] + 3[x.sup.2][y.sup.2] + (6n - 14)[x.sup.2][y.sup.3] + (3 (n - 2)(n - 3)/2)[x.sup.3][y.sup.3]. (25)

Then,

[mathematical expression not reproducible]. (26)

Now, we have

[mathematical expression not reproducible]. (27)

Corollary 5. Let O[P.sub.n] be the swapped network; then,

[mathematical expression not reproducible]. (28)

Proof. Let

f (x, y) = M (G; x, y) = 2x[y.sub.3] + 3[x.sup.2][y.sup.2] + (6n - 14)[x.sup.2][y.sup.3] + (3(n - 2)(n - 3)/2)[x.sup.3][y.sup.3]. (29)

Then,

[mathematical expression not reproducible]. (30)

Now, we have

[mathematical expression not reproducible]. (31)

Corollary 6. Let O[P.sub.n] be the swapped network; then,

SSD (O[P.sub.n]) = 3[n.sup.2] - 2n + 1/3. (32)

Proof. Let

f (x, y) = M (G; x, y) = 2x[y.sup.3] + 3[x.sup.2][y.sup.2] + (6n - 14)[x.sup.2][y.sup.3] + (3 (n - 2)(n - 3)/2)[x.sup.3][y.sup.3]. (33)

Then,

[mathematical expression not reproducible]. (34)

Now, we have

SSD (O[P.sub.n]) = ([D.sub.x][S.sub.y]f + [S.sub.x][D.sub.y]f)[|.sub.x=y=1] = 3[n.sup.2] - 2n + 1/3. (35)

Corollary 7. Let O[P.sub.n] be the swapped network; then,

H(O[P.sub.n]) = 2([n.sup.2]/4 - n/20 - 1/20). (36)

Proof. Let

f(x, y) = M (G; x,y) = 2x[y.sup.3] + 3[x.sup.2][y.sup.2] + (6n - 14)[x.sup.2][y.sup.3] + (3(n - 2)(n - 3)/2)[x.sup.3][y.sup.3]. (37)

Then,

J(f(x, y)) = 5[x.sup.4] + (6n - 14)[x.sup.5] + (3(n - 2)(n -3)/2)[x.sup.6], 2[S.sub.x]J(f(x, y)) = (5[x.sup.4]/4 + (6n - 14)[x.sup.5]/5 + (n - 2)(n - 3)[x.sup.6]/4). (38)

Now, we have

H(O[P.sub.n]) = 2[S.sub.x]Jf[|.sub.x=1] = 2([n.sup.2]/4 - n/20 - 1/20). (39)

Corollary 8. Let O[P.sub.n] be the swapped network; then,

I(O[P.sub.n]) = 9[n.sup.2]/4 81n/20 - 129/5. (40)

Proof. Let

f(x, y) = M (G; x, y) = 2x[y.sup.3]0 + 3[x.sup.2][y.sup.2] + (6n - 14)[x.sup.2][y.sup.3] + (3(n - 2)(n - 3)/2)[x.sup.3][y.sup.3]. (41)

Then,

[mathematical expression not reproducible]. (42)

Now, we have

I(O[P.sub.n]) = [S.sub.x]J[D.sub.x][D.sub.y]f[|.sub.x=i] = 9[n.sup.2]/4 81n/20 129/5. (43)

Corollary 9. Let O[P.sub.n] be the swapped network; then, A(O[P.sub.n]) = 729[n.sub.2]/64 573n/64 - 413/32. (44)

Proof. Let

f(x, y) = M (G; x, y) = 2x[y.sub.3] + 3[x.sub.2][y.sup.2] + (6n - 14)[x.sup.2][y.sup.3] + (3 (n - 2)(n - 3)/2)[x.sup.3][y.sup.3]. (45)

Then,

[mathematical expression not reproducible]. (46)

Now, we have

A(O[P.sub.n]) = [S.sup.3.sub.x][Q.sub.-2]J[D.sup.3.sub.x][D.sup.3.sub.y]f[|.sub.x=1] = 729[n.sup.2]/64 - 573n/64 - 413/32. (47)

5.2. Results for (O[R.sub.k])

Theorem 2. Let O[R.sub.k] be the swapped network; then,

M (G; x, y) = nk[x.sup.k] [y.sup.k+1] + ([n.sup.2] (k + 1) - n(1 + 2k)/2)[x.sub.k+1][y.sub.k+1]. (48)

Proof. The O[R.sub.k] network has the following two types of edges based on the degree of end vertices:

[E.sub.{k,k+1}](O[R.sub.k]) = {uv [member of] E(O[R.sub.k]) : [d.sub.u] = k, [d.sub.v] = k + 1}, [E.sub.{k+1,k+1}] (O[R.sub.k]) = {uv [member of] E(O[R.sub.k]) : [d.sub.u] = k + 1, [d.sub.v] = k + 1}, (49)

such that

[absolute value of [E.sub.{k,k+1}] O[R.sub.k])] = nk, [absolute value of [E.sub.{k+1,k+1}] O[R.sub.k])] = [n.sup.2] (k + 1) - n(1 + 2k)/2. (50)

Now, from the definition of M-polynomial, we have

[mathematical expression not reproducible]. (51)

Corollary 10. Let ([degrees]Rk) be the swapped network; then, [M.sub.1](O[R.sub.k]) = 2n[k.sup.2] + nk + [n.sup.2] [(k + 1).sup.2] - n(k + 1)(1 + 2k). (52)

Proof. Let

f(x, y) = M (G; x, y) = [nkx.sup.k][y.sup.k+1] + ([n.sup.2] (k + 1) - n(1 + 2k)/2) [x.sup.k+1] [y.sup.k+1]. (53)

Then,

[mathematical expression not reproducible]. (54)

Now, we have

[M.sub.1] (O[R.sub.k]) = [D.sub.x]f + [D.sub.y]f[|.sub.x=y=1] = 2n[k.sup.2] + nk + [n.sup.2] [(k + 1).sup.2] - n(k + 1)(1 + 2k). (55)

Corollary 11. Let (O[R.sub.k]) be the swapped network; then,

[M.sub.2] (O[R.sub.k]) = n[k.sup.3] + n[k.sup.2] + [n.sup.2][(k + 1).sup.3]/2 - n[(k + 1).sup.2] (1 + 2k)/2. (56)

Proof. Let

f(x, y) = M (G; x, y) = [nkx.sup.k] [y.sup.k+1] + ([n.sup.2] (k + 1) - n(1 + 2k)/2) [x.sup.k+1] [y.sup.k+1]. (57)

Then,

[mathematical expression not reproducible]. (58)

Now, we have

[M.sub.2] (O[R.sub.k]) = [D.sub.x][D.sub.y]f [|.sub.x=y=1] = [nk.sup.3] + [nk.sup.2] + [n.sup.2][(k + 1).sup.3]/2 n[(k + 1).sup.2] (1 + 2k)/2. (59)

Corollary 12. Let ([degrees]Rk) be the swapped network; then,

[sup.m][M.sub.2](O[R.sub.k]) = nk/([k.sup.2] + k) + [n.sup.2]/(2k + 2) - n(1 + 2k)/2[(k + 1).sup.2]. (60)

Proof. Let

f(x,y) = M(G; x,y) = [nkx.sup.k] [y.sup.k+1] + ([n.sup.2] (k + 1) - n(1 + 2k)/2) [x.sup.k+1] [y.sup.k+1]. (61)

Then,

[mathematical expression not reproducible]. (62)

Now, we have

[sup.m][M.sub.2](O[R.sub.k]) = [S.sub.x][S.sub.y]f[|.sub.x=y=1] = nk/([k.sup.2] + k) + [n.sup.2]/(2k + 2) - n(1 + 2k)/2[(k + 1).sup.2]. (63)

Corollary 13. Let (O[R.sub.k]) be the swapped network; then,

[R.sub.[alpha]]((O[R.sub.k]) = [nk.sup.3[alpha]] + [nk.sup.2[alpha]] + [(k + 1).sup.2[alpha]] ([n.sup.2](k + 1)- n(1 + 2k)/[2.sup.[alpha]]) [x.sup.k+1] [y.sup.k+1]). (64)

Proof. Let

f(x, y) = M (G; x, y) = [nkx.sup.k][y.sup.k+1] + ([n.sup.2] (k + 1) - n(1 + 2k)/2) [x.sup.k+1] [y.sup.k+1]. (65)

Then,

[mathematical expression not reproducible]. (66)

Now, we have

[R.sub.[alpha]]((O[R.sub.k]) == [D.sup.[alpha].sub.x] + [D.sup.[alpha].sub.y]f|[sub.x=y=1] = [nk.sup.3[alpha]] [nk.sup.2[alpha]] + [(k + 1).sup.2[alpha]] ([n.sup.2] (k + 1) - n(1 + 2k)/[2.sup.[alpha]]) [x.sup.k+1] [y.sup.k+1]). (67)

Corollary 14. Let (O[R.sub.k]) be the swapped network; then,

R[R.sub.[alpha]]((O[R.sub.k]) = n/[(k + 1).sup.[alpha]] + ([n.sup.2](k + 1) - n(1 + 2k)/[(k + 1).sup.2[alpha]][2.sup.[alpha]])). (68)

Proof. Let

f(x, y) = M (G; x, y) = [nkx.sub.k][y.sup.k+1] + ([n.sup.2] (k + 1) - n(1 + 2k)/2) [x.sup.k+1] [y.sup.k+1]. (69)

Then,

[mathematical expression not reproducible]. (70)

Now, we have

R[R.sub.[alpha]]((O[R.sub.k]) == [S.sup.[alpha].sub.x] + [S.sup.[alpha].sub.y]f|[sub.x=y=1] = n/[(k + 1).sup.[alpha]] + ([n.sup.2](k + 1) - n(1 + 2k)/(k + 1).sup.2[alpha]][2.sup.[alpha]])). (71)

Corollary 15. Let (O[R.sub.k]) be the swapped network; then, SSD(O[R.sub.k]) = n[k.sup.2]/(k + 1) + nk + n + [n.sup.2] (k + 1) - n(1 + 2k). (72)

Proof. Let

f(x,y) = M(G; x,y) = [nkx.sup.k] [y.sup.k+1] + ([n.sup.2] (k + 1) - n(1 + 2k)/2) [x.sup.k+1] [y.sup.k+1]. (73)

Then,

[mathematical expression not reproducible]. (74)

Now, we have

SSD(O[R.sub.k]) = ([D.sub.x][S.sub.y]f + [S.sub.x][D.sub.y]f)[|.sub.x=y=1] = n[k.sup.2]/(k + 1) + nk + n + [n.sup.2] (k + 1) - n(1 + 2k). (75)

Corollary 16. Let (O[R.sub.k]) be the swapped network; then,

H(O[R.sub.k]) = 2(nk/2k + 1 + [n.sup.2] (k + 1)/4k + 4 - n(1 + 2k)/4k + 4). (76)

Proof. Let

f (x, y) = M (G; x, y) = [nkx.sup.k] [y.sup.k+1] + ([n.sup.2] (k + 1) - n(1 + 2k)/2) [x.sup.k+1] [y.sup.k+1]. (77)

Then,

[mathematical expression not reproducible]. (78)

Now, we have

H(O[R.sub.k]) = 2[S.sub.x]Jf[|.sub.x=1] = 2(nk/2k + 1 + [n.sup.2] (k + 1)/4k + 4 - n(1 + 2k)/4k + 4). (79)

Corollary 17. Let (ORk) be the swapped network; then,

I(O[R.sub.k]) = n[k.sup.3]/2k + 1 + n[k.sup.2]/2k + 1 + [(k + 1).sup.2]([n.sup.2] (k + 1) - n(1 + 2k)/4k + 4). (80)

Proof. Let

f (x, y) = M (G; x, y) = [nkx.sup.k] [y.sup.k+1] + ([n.sup.2] (k + 1) - n(1 + 2k)/2) [x.sup.k+1] [y.sup.k+1]. (81)

Then,

[mathematical expression not reproducible]. (82)

Now, we have

[mathematical expression not reproducible]. (83)

Corollary 18. Let (O[R.sub.k]) be the swapped network; then,

[mathematical expression not reproducible]. (84)

Proof. Let

f (x, y) = M (G; x, y) = [nkx.sup.k] [y.sup.k+1] + ([n.sup.2] (k + 1) - n(1 + 2k)/2) [x.sup.k+1] [y.sup.k+1]. (85)

Then,

[mathematical expression not reproducible]. (86)

Now, we have

[mathematical expression not reproducible]. (87)

6. Conclusion

In this paper, our focus is on swapped interconnection networks that allow systematic construction of large, scalable, modular, and robust parallel architectures, while maintaining many desirable attributes of the underlying basis network comprising its clusters. We have computed several TIs of underlined networks. Firstly, we computed Mpolynomials of understudy networks, and then we recovered Zagreb indices, Randic indices, and some other indices [49-52].

https://doi.org/10.1155/2019/4291943

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they no conflicts of interest.

Authors' Contributions

All authors contributed equally to this study.

Acknowledgments

This study was partially supported by the Natural Science Foundation of Anhui Province (no. KJ2018A059).

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Hai-Xia Li, (1) Sarfaraz Ahmad, (2) and Iftikhar Ahmad (2,3)

(1) Department of General Education, Anhui Xinhua University, Hefei 230088, China

(2) Department of Mathematics, COMSATS University of Islamabad, Lahore Campus, Lahore 54000, Pakistan

(3) Department of Mathematics, Riphah International University, Lahore Campus, Lahore 54000, Pakistan

Correspondence should be addressed to Iftikhar Ahmad; iffi6301@gmail.com

Received 2 September 2019; Accepted 10 October 2019; Published 7 November 2019

Guest Editor: Shaohui Wang

Caption: Figure 1: [mathematical expression not reproducible].

Caption: Figure 2: O[P.sub.7].
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Title Annotation:Research Article
Author:Li, Hai-Xia; Ahmad, Sarfaraz; Ahmad, Iftikhar
Publication:Journal of Chemistry
Geographic Code:4EXCR
Date:Nov 1, 2019
Words:5347
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