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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


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Volume 16, Issue 1

Issues

Volume 13 (2015)

Super (a, d)-H-antimagic labeling of subdivided graphs

Amir Taimur / Muhammad Numan / Gohar Ali / Adeela Mumtaz / Andrea Semaničová-Feňovčíková
Published Online: 2018-06-22 | DOI: https://doi.org/10.1515/math-2018-0062

Abstract

A simple graph G = (V, E) admits an H-covering, if every edge in E(G) belongs to a subgraph of G isomorphic to H. A graph G admitting an H-covering is called an (a, d)-H-antimagic if there exists a bijective function f : V(G) ∪ E(G) → {1, 2, …, |V(G)| + |E(G)|} such that for all subgraphs H′ isomorphic to H the sums ∑vV(H′)f(v) + ∑eE(H′)f(e) form an arithmetic sequence {a, a + d, …, a + (t − 1)d}, where a > 0 and d ≥ 0 are integers and t is the number of all subgraphs of G isomorphic to H. Moreover, if the vertices are labeled with numbers 1, 2, …, |V(G)| the graph is called super. In this paper we deal with super cycle-antimagicness of subdivided graphs. We also prove that the subdivided wheel admits an (a, d)-cycle-antimagic labeling for some d.

Keywords: H-covering; (Super) (a, d)-H-antimagic labeling; Subdivided graph; Subdivided wheel

MSC 2010: 05C78

1 Introduction

Let G = (V, E) be a finite simple graph with the vertex set V(G) and the edge set E(G). An edge-covering of G is a family of subgraphs H1, H2, …, Ht such that each edge of E belongs to at least one of the subgraphs Hi, i = 1, 2, …, t. Then it is said that G admits an (H1, H2, …, Ht)-(edge) covering. If every subgraph Hi is isomorphic to a given graph H, then the graph G admits an H-covering. A bijective function f : V(G) ∪ E(G) → {1,2, …, |V(G)| + |E(G)| } is an (a, d)-H-antimagic labeling of a graph G admitting an H-covering whenever, for all subgraphs H′ isomorphic to H, the H′-weights

wtf(H)=vV(H)f(v)+eE(H)f(e)

form an arithmetic progression a, a + d, …, a + (t − 1)d, where a > 0 and d ≥ 0 are two integers, and t is the number of all subgraphs of G isomorphic to H. Such a labeling is called super if the smallest possible labels appear on the vertices. A graph that admits a (super) (a, d)-H-antimagic labeling is called (super) (a, d)-H-antimagic. For d = 0 it is called H-magic and H-supermagic, respectively.

The H-(super)magic labelings were first studied by Gutiérrez and Lladó [1] as an extension of the edge-magic and super edge-magic labelings introduced by Kotzig and Rosa [2] and Enomoto, Lladó, Nakamigawa and Ringel [3], respectively. In [1] are considered star-(super)magic and path-(super)magic labelings of some connected graphs and it is proved that the path Pn and the cycle Cn are Ph-supermagic for some h. Lladó and Moragas [4] studied the cycle-(super)magic behavior of several classes of connected graphs. They proved that wheels, windmills, books and prisms are Ch-magic for some h. Maryati, Salman, Baskoro, Ryan and Miller [5] and also Salman, Ngurah and Izzati [6] proved that certain families of trees are path-supermagic. Ngurah, Salman and Susilowati [7] proved that chains, wheels, triangles, ladders and grids are cycle-supermagic. Maryati, Salman and Baskoro [8] investigated the G-supermagicness of a disjoint union of c copies of a graph G and showed that the disjoint union of any paths is cPh-supermagic for some c and h.

The (a, d)-H-antimagic labeling was introduced by Inayah, Salman and Simanjuntak [9]. In [10] there are investigated the super (a, d)-H-antimagic labelings for some shackles of a connected graph H. In [11] was proved that wheels are cycle-antimagic. In [12] it was shoved that if a graph G admits a (super) (a, d)-H-antimagic labeling, where d = |E(H)| − |V(H)|, then the disjoint union of m copies of the graph G, denoted by mG, admits a (super) (b, d)-H-antimagic labeling as well. Rizvi, et al. [13] proved the disjoint union of isomorphic copies of fans, triangular ladders, ladders, wheels, and graphs obtained by joining a star K1,n with K1, and also disjoint union of non-isomorphic copies of ladders and fans are cycle-supermagic.

In this paper we will discuss a super cycle-atimagicness of subdivided graphs. We show that the property to be super (a, d)-H-antimagic is hereditary according to the operation of subdivision of edges. We prove that if a graph G is super cycle-antimagic then the subdivided graph S(G) also admits a super cycle-antimagic labeling. Moreover, we show that the subdivided wheel is super (a, d)-cycle-antimagic for wide range of differences.

2 Subdivided graphs

Let us consider the graph S(G) obtained by subdividing some edges of a graph G, thus by inserting some new vertices to the original graph G. Equivalently, the graph S(G) can by obtained from G by replacing some edges of G by paths. The topic of subdivided graphs has been widely studied in recent years, for example see [14].

Let G be a graph admitting H-covering given by t subgraphs H1, H2, …, Ht isomorphic to H. Let us consider the subgraphs SG(Hi), i = 1, 2, …, t, corresponding to Hi in S(G). If these subgraphs are all isomorphic to a graph, let us denote it by the symbol SG(H), then the graph S(G) admits SG(H)-covering.

The next theorem shows that the property of being super (a, d)-H-antimagic is hereditary according to the operation of subdivision of edges.

Theorem 2.1

Let G be a super (a, d)-H-antimagic graph and let Hi, i = 1, 2, …, t, be all subgraphs of G isomorphic to H. If SG(Hi), i = 1, 2, …, t, are all subgraphs of S(G) isomorphic to SG(H) then the graph S(G) is a super (b, d)-S(H)-antimagic graph.

Proof

Let G be a super (a, d)-H-antimagic graph and let Hi, i = 1, 2, …, t, be all subgraphs of G isomorphic to H. Let f be a super (a, d)-H-antimagic labeling of G, thus f : V(G) ∪ E(G) → {1, 2, …, |V(G)| + |E(G)|} such that the vertices of G are labeled with numbers 1, 2, …, |V(G)| and the weights of subgraphs Hi, i = 1, 2, …, t,

wtf(Hi)=vV(Hi)f(v)+eE(Hi)f(e)

form an arithmetic progression a, a + d, …, a + (t − 1)d, where a > 0 and d ≥ 0 are two integers, i.e.,

{wtf(Hi):i=1,2,,t}={a,a+d,,a+(t1)d}.(1)

Let us consider the graph S(G) obtained from G by inserting p new vertices, say v1, v2, …, vp, to the edges of G. Let SG(Hi), i = 1, 2, …, t, be all subgraphs of S(G) isomorphic to SG(H). Then S(G) admits the SG(H)-covering. Let r denote the number of new vertices inserted to every subgraph SG(Hi), i = 1, 2, …, t.

We define a labeling g of S(G) in the following way

g(v)={f(v),ifvV(G),|V(G)|+j,ifv=vj,j=1,2,,p.

Evidently, the vertices of S(G) are labeled with distinct numbers 1, 2, …, |V(G)| + p.

Let us choose an orientation of edges in G. According to this orientation we orient the edges in S(G). To an arc uv in G there will correspond the oriented path Puv with initial vertex u and terminal vertex v in S(G). The arcs of S(G) we label such that

g(uw)={f(uv)+p,ifuV(G)anduwis an arc onPuv,|V(G)|+|E(G)|+2p+1j,ifu=vj,j=1,2,,p.

The edges are labeled with distinct numbers from the set |V(G)| + p + 1, |V(G)| + p + 2, …, |V(G)| + |E(G)| + 2p. Now we evaluate the weights of subgraphs SG(Hi), i = 1, 2, …, t, under the labeling g. Immediately using the structure of the subgraph SG(Hi) and the definition of the labeling g we get

wtg(SG(Hi))=vV(G(Hi))g(v)+eE(G(Hi))g(e)=vV(Hi)g(v)+vjV(G(Hi))g(vj)+eE(P(uv)),uvE(Hi)g(e)=vV(Hi)f(v)+vjV(G(Hi))(|V(G)|+j)+eE(Hi)(f(e)+p)+vjV(G(Hi))(|V(G)|+|E(G)|+2p+1j)=vV(Hi)f(v)+eE(Hi)f(e)+|E(Hi)|p+(2|V(G)|+|E(G)|+2p+1)r=wtf(Hi)+|E(Hi)|p+(2|V(G)|+|E(G)|+2p+1)r.

As |E(Hi)| = |E(H)| for i = 1, 2, …, t we obtain that the weights of SG(Hi) depend on the weights of Hi which form an arithmetic sequence with a difference d, see (1). This implies that the set of weights SG(Hi) also forms an arithmetic sequences with the difference d and the initial term a + |E(H)|p + (2|V(G)| + |E(G)| + 2p + 1)r. This concludes the proof. □

Combining Theorem 2.1 with some results on (a, d)-cycle-antimagic graphs we immediately obtain new classes of graphs that are (b, d)-cycle-antimagic. Note, that it is not needed to consider only regular subdivisions of graphs.

3 Subdivided wheels

A wheel Wn is a graph obtained by joining a single vertex to all vertices of a cycle on n vertices. The vertex of degree n is called the central vertex, or the hub vertex, and the remaining vertices are called the rim vertices. The edges adjacent to the central vertex are called spokes and the remaining edges are called rim edges. Let us denote by the symbol Wn(r, s) the graph obtained by inserting r, r ≥ 0, new vertices to every rim edge and s, s ≥ 0, new vertices to every spoke in the wheel Wn. Note, that the graph isomorphic to subdivided wheel Wn(r, 0) is also known as the Jahangir graph Jn,r+1.

In [11] it was proved that wheels are cycle-antimagic.

Theorem 3.1

([11]). Let k and n ≥ 3 be positive integers. The wheel Wn is super (a, 1)-Ck-antimagic for every k = 3, 4, …, n − 1, n + 1.

Immediately using Theorem 2.1 we obtain that subdivided wheels admit cycle-antimagic labeling with difference 1.

Corollary 3.2

Let k, n ≥ 3, r ≥ 0, s ≥ 0 be integers. The subdivided wheel Wn(r, s) is super (a, 1)-Ck+(k−2)r+2s-antimagic for every k = 3, 4, …, n − 1, n + 1.

In the next theorem we will deal with the cycle-antimagicness of the subdivided wheel Wn(1, 1). We prove that this graph admits a super (a, d)-C6-antimagic labeling for d ∈ {0, 1, …,5}.

Theorem 3.3

The subdivided wheel Wn(1, 1), n ≥ 3, is super (a, d)-C6-antimagic for d ∈ {0, 1, …,5}.

Proof

Let us denote the vertices and edges of Wn(1, 1) such that

V(Wn(1,1))={c,vi,ui,wi:i=1,2,,n},E(Wn(1,1))={cwi,wivi,viui,uivi+1:i=1,2,,n},

where the indices are taken modulo n.

For d = 1 the result follows from Corollary 3.2. For d ∈ {0, 2,3, 4,5} we define a total labeling gd:V(Wn(1, 1)) ∪ E(Wn(1, 1)) → {1, 2, …, 7n + 1} in the following way.

gd(c)=1,ford=0,2,3,4,5,g0(wi)={2,fori=1,n+3i,for2in,g0(ui)=3n+22i,for1in,g0(vi)={n1+2i,for2in,3n+1,fori=1,g0(cwi)=3n+1+i,for1in,g0(wivi)={5n+1+i,for1in1,5n+1,fori=n,g0(viui)={6n+3i,for2in,5n+2,fori=1,g0(uivi+1)={6n+3+i,for1in2,5n+3+i,forn1in,g2(wi)=3n+2i,for1in,g2(ui)=1+2i,for1in,g2(vi)=2i,for1in,g2(cwi)=4n+2i,for1in,g2(wivi)=5n+2i,for1in,g2(viui)={5n+2+i,for1in1,5n+2,fori=n,g2(uivi+1)=6n+1+i,for1in,g3(wi)={2,fori=1,n+3i,for2in,g3(ui)=3n+2i,for1in,g3(vi)={2n+1,fori=1,n+i,for2in,g3(cwi)={4n+1,fori=1,3n+i,for2in,g3(wivi)={4n+2,fori=1,5n+3i,for2in,g3(viui)=5n+2i,for1in,g3(uivi+1)=5n+2i+1,for1in,g4(wi)={2,fori=1,n+3i,for2in,g4(ui)=n+2i,for1in,g4(vi)={3n+1,fori=1,n1+2i,for2in,g4(cwi)=3n+1+i,for1in,g4(wivi)={5n+1i,for1in1,5n+1,fori=n,g4(viui)={5n+2,fori=1,6n+3i,for2in,g4(uivi+1)={6n+3+i,for1in2,5n+3+i,forn1in,g5(wi)={2,fori=1,n+3i,for2in,g5(vi)={2n+1,fori=1,n+i,for2in,g5(ui)=2n+1+i,for1in,g5(cwi)={3n+i,for2in,4n+1,fori=1,g5(wivi)={4n+2,fori=1,5n+3i,for2in,g5(viui)=5n+2i,for1in,g5(uivi+1)=5n+2i+1,for1in.

We denote by the symbol C6i, 1 ≤ in, the 6-cycle such that C6i = cwiuiviui+1 wi+1, where the index i is taken modulo n. Under the labeling gd, the weights of C6i are as follows.

wtg(C6i)=g(c)+g(wi)+g(ui)+g(vi)+g(ui+1)+g(wi+1)+g(cwi)+g(wiui)+g(uivi)+g(viui+1)+g(ui+1wi+1)+g(wi+1c).

It is a simple mathematical exercise to prove that for every i, 1 ≤ in, the 6-cycle-weights are:

wtg0(C6i)=35n+18,for1in,wtg2(C6i)={36n+17,fori=1,34n+15+2i,for2in,wtg3(C6i)=33n+16+3i,for1in,wtg4(C6i)=33n+16+4i,for1in,wtg5(C6i)=32n+15+5i,for1in.

Hence the weights of cycles C6 form an arithmetic sequence with differences d = 0, 2,3, 4,5, respectively. This concludes the proof. □

Combining Theorem 2.1 and Theorem 3.3 we immediately obtain the following result.

Theorem 3.4

The subdivided wheel Wn(r, s), n ≥ 3, r ≥ 1 and s ≥ 1 is super (a, d)-Cr+2s+3-antimagic for d ∈ {0, 1, 2, 3, 4, 5}.

In the next section we will deal with the subdivided wheel Wn(r, 0), n ≥ 3, r ≥ 1. Let us denote the vertices and the edges of Wn(r, 0) such that

V(Wn(r,0))={c,vi,uji:1in,1jr}E(Wn(r,0))={cvi,viu1i:1in}{urivi+11in1}{urnv1}{ujiuj+1i:1in,1jr1}.

The subdivided wheel Wn(r, 0), n ≥ 3, r ≥ 1, has n vertices of degree 3, nr vertices of degree 2 and one vertex of degree n. The size of Wn(r, 0) is n(r + 2).

The subdivided wheel Wn(r, 0) admits the Cr+3-covering consisting of n cycles Cr+3. Let us denote these cycles by the symbols Cr+3i, i = 1, 2, …, n, such that Cr+3i=cviu1iu2iurivi+1.

The following theorem shows the existence of a super (a, d)-Cr+3-antimagic labeling for Wn(r, 0) for every odd difference form 1 up to 2r − 3.

Theorem 3.5

The subdivided wheel Wn(r, 0), n ≥ 3, r ≥ 1, is super (a, d)-Cr+3-antimagic for d = 1 when r = 1 and for d ≡ 1 (mod 2), 1 ≤ d ≤ 2r − 3 when r ≥ 1.

Proof

For r = 1 the result follows from Corollary 3.2. Let r ≥ 2 and let d be an odd positive integer, 1 ≤ d ≤ 2r − 3. Let fd:V(Wn(r, 0)) ∪ E(Wn(r, 0)) → {1, 2, …, n(2r + 3) + 1} be a labeling of Wn(r, 0), n ≥ 3, r ≥ 2, defined in the following way.

fd(c)=1,fd(vi)=1+i,for1in,fd(uji)=jn+1+i,for1in,1jr,fd(urnv1)=2nr+3n+1,fd(urivi+1)=2nr+2n+1+i,for1in1,fd(ujiuj+1i)=nr+3n+jn+2i,for1in,1jr(d+1)/2,fd(ujiuj+1i)=nr+2n+1+i+jn,for1in,r(d1)/2jr1,fd(viu1i)=nr+3n+2i,for1in,fd(cvi)=nr+2n+2i,for1in.

It is easy to see that fd is a bijection as

fd(c)=1,{fd(vi):1in}={2,3,,n+1},{fd(uji):1in,1jr}={n+2,n+3,,nr+n+1},fd(urnv1)=2nr+3n+1,{fd(urivi+1):1in1}={2nr+2n+2,2nr+2n+3,,2nr+3n},{fd(ujiuj+1i):1in,1jr(d+1)/2}7={nr+3n+2,nr+3n+3,,2nr+3nn(d1)/2+1},{fd(ujiuj+1i):1in,r(d1)/2jr1}={2nr+3nn(d1)/2+2,2nr+3nn(d1)/2+3,,2nr+2n+1},{fd(viu1i):1in}={nr+2n+2,nr+2n+3,,nr+3n+1},{fd(cvi):1in}={nr+n+2,nr+n+3,,nr+2n+1}.

Under the labeling fd the weights of cycles Cr+3i, i = 1, 2, …, n − 1, are as follows.

wtfd(Cr+3i)=vV(Cr+3i)fd(v)+eE(Cr+3i)fd(e)=fd(c)+fd(vi)+fd(vi+1)+j=1rfd(uji)+fd(cvi)+fd(cvi+1)+fd(viu1i)+j=1r1fd(ujiuj+1i)+fd(urivi+1)=1+(1+i)+(1+(i+1))+j=1r(jn+1+i)+(nr+2n+2i)+(nr+2n+2(i+1))+(nr+3n+2i)+j=1r(d+1)/2(nr+3n+jn+2i)+j=r(d1)/2r1(nr+2n+1+i+jn)+(2nr+2n+1+i)=2nr2+7nr+7n+3r+9(d+1)(n+1)2+di.

Moreover, the weight of the cycle Cr+3n we get

wtfd(Cr+3n)=vV(Cr+3n)fd(v)+eE(Cr+3n)fd(e)=fd(c)+fd(vn)+fd(v1)+j=1rfd(ujn)+fd(cvn)+fd(cv1)+fd(vnu1n)+j=1r1fd(ujnuj+1n)+fd(urnv1)=1+(1+n)+2+j=1r(jn+1+n)+(nr+n+2)+(nr+2n+1)+(nr+2n+2)+j=1r(d+1)/2(nr+2n+jn+2)+j=r(d1)/2r1(nr+3n+1+jn)+(2nr+3n+1)=2nr2+7nr+6n+3r+9(d+1)(n+1)2.

This proves that fd is a super (a, d)-Cr+3-antimagic labeling of Wn(r, 0) for d ≡ 1 (mod 2), 1 ≤ d ≤ 2r − 3 and a = 2nr2 + 7nr + 6n + 3r + 9 − (d + 1)(n + 1)/2. □

In the next theorem we prove that the graph Wn(r, 0) admits super (a, d)-Cr+3-antimagic labelings also for even differences.

Theorem 3.6

The subdivided wheel Wn(r, 0), r ≥ 1, is super (a, d)-Cr+3-antimagic for d = 0 when r = 1, n ≥ 5 and for d ≡ 0 (mod 2), 0 ≤ d ≤ 2r − 4 when r ≥ 2, n ≥ 3.

Proof

Lladó and Moragas [4] proved that the wheel Wn, n ≥ 5 odd, is (a, 0)-C3-antimagic. From Corollary 3.2 we obtain that Wn(r, 0), n ≥ 5, r ≥ 1, is super (b, 0)-Cr+3-antimagic.

Let r ≥ 2, n ≥ 3 be positive integers. Let d be an even integer, 0 ≤ d ≤ 2r − 4. Let fd:V(Wn(r, 0)) ∪ E(Wn(r, 0)) → {1, 2, …, n(2r + 3) + 1} be a labeling of Wn(r, 0), n ≥ 3, r ≥ 1, defined in the following way.

gd(c)=1,gd(vi)=2i,for1in,gd(u1i)=2n2i+3,for1in,gd(uji)=jn+1+i,for1in,2jr,gd(urnv1)=nr+n+2,gd(urivi+1)=nr+2n+2i,for1in1,gd(ujiuj+1i)=2nr+n+1+ijn,for1in,1j(1+d/2),gd(ujiuj+1i)=2nr+2njn+2i,for1in,(2+d/2)jr1,gd(viu1i)=2nr+2n+1i,for1in1,gd(vnu1n)=2nr+2n+1,gd(cvi)=2nr+3n+2i,for1in.

The labeling gd is a bijection. Under the labeling gd the weights of cycles Cr+3i, i = 1, 2, …, n − 1, are the following.

wtgd(Cr+3i)=vV(Cr+3i)gd(v)+eE(Cr+3i)gd(e)=gd(c)+gd(vi)+gd(vi+1)+j=1rgd(uji)+gd(cvi)+gd(cvi+1)+gd(viu1i)+j=1r1gd(ujiuj+1i)+gd(urivi+1)=1+2i+2(i+1)+(2n2i+3)+j=2r(jn+1+i)+(2nr+3n+2i)+(2nr+3n+2(i+1))+(2nr+2n+1i)+j=11+d/2(2nr+n+1+ijn)+j=2+d/2r1(2nr+2njn+2i)+(nr+2n+2i)=2nr2+8nr+8n+3r+8d(n+1)2+di.

For the weight of the cycle Cr+3n we obtain:

wtgd(Cr+3n)=vV(Cr+3n)gd(v)+eE(Cr+3n)gd(e)=gd(c)+gd(vn)+gd(v1)+j=1rgd(ujn)+gd(cvn)+gd(cv1)+gd(vnu1n)+j=1r1gd(ujnuj+1n)+gd(urnv1)=1+2n+2+3+j=2r(jn+1+n)+(2nr+2n+2)+(2nr+3n+1)+(2nr+2n+1)+j=11+d/2(2nr+2n+1jn)+j=2+d/2r1(2nr+n+2jn)+(nr+n+2)=2nr2+8nr+8n+3r+8+nd2d2.

We showed that gd is a super (a, d)-Cr+3-antimagic labeling of Wn(r, 0) for d ≡ 0 (mod 2), 0 ≤ d ≤ 2r − 4 and a = 2nr2+8nr + 8n + 3r + 8 − dn/2 + d/2. □

Combining Theorem 3.5 and Theorem 3.6 we immediately obtain that the subdivided wheel Wn(r, 0), n ≥ 5, is cycle-antimagic for wide range of differences.

Theorem 3.7

The subdivided wheel Wn(r, 0), n ≥ 5, is super (a, d)-Cr+3-antimagic for 0 ≤ d ≤ 1 when r = 1 and for 0 ≤ d ≤ 2r − 3 when r ≥ 2.

Moreover, using Theorem 2.1, we can extend this result also for subdivided wheels in which not only rim edges but also spokes are subdivided.

Theorem 3.8

The subdivided wheel Wn(r, s), n ≥ 5, r ≥ 1, s ≥ 0, is super (a, d)-Cr+2s+3-antimagic for 0 ≤ d ≤ 1 when r = 1 and for 0 ≤ d ≤ 2r − 3 when r ≥ 2.

4 Conclusion

In the present paper we showed that the property to be super (a, d)-H-antimagic is hereditary according to the operation of subdivision of edges. We proved that if a graph G is super cycle-antimagic then the subdivided graph S(G) also admits a super cycle-antimagic labeling.

This indicates that it is important to study the antimagic properties of graphs with simple structures which allows us to get result for large graphs. Recently, large graphs have attracted a lot of attention, see [15]. However, the interesting question is whether, for a given graph, it is possible to extend the set of differences also for cases not covered by the general result. It means to find a difference d such that the subdivided graph S(G) is super cycle-antimagic with the difference d but the corresponding graph G is not.

Another interesting directions for further investigation is to deal with the non-uniform subdivision and to find another graph operations that are hereditary according to being cycle-antimagic, or in general H-antimagic.

Acknowledgement

The research for this article was supported by APVV-15-0116 and by VEGA 1/0233/18.

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About the article

Received: 2017-11-07

Accepted: 2018-05-09

Published Online: 2018-06-22


Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 688–697, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0062.

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© 2018 Taimur et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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