IJMMSInternational Journal of Mathematics and Mathematical Sciences1687-04250161-1712Hindawi10.1155/2020/78128127812812Research ArticleOn the Construction of the Reflexive Vertex k-Labeling of Any Graph with Pendant Vertexhttps://orcid.org/0000-0001-7508-0760AgustinI. H.12https://orcid.org/0000-0003-2292-8443UtoyoM. I.2https://orcid.org/0000-0003-0575-3039Dafik13VenkatachalamM.4Surahmat5KriegAloys1CGANT-University of JemberJemberIndonesiaunej.ac.id2Department of MathematicsAirlangga UniversitySurabayaIndonesiaunair.ac.id3Department of Mathematics EducationUniversity of JemberJemberIndonesiaunej.ac.id4Department of MathematicsKongunadu Arts and Science CollegeCoimbatoreIndiakongunaducollege.ac.in5Department of Mathematics EducationUniversitas Islam MalangMalangIndonesiaunisma.ac.id202011020202020060520202307202011020202020Copyright © 2020 I. H. Agustin et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A total k-labeling is a function fe from the edge set to first natural number ke and a function fv from the vertex set to non negative even number up to 2kv, where k=maxke,2kv. A vertex irregular reflexivek-labeling of a simple, undirected, and finite graph G is total k-labeling, if for every two different vertices x and x of G, wtxwtx, where wtx=fvx+ΣxyEGfexy. The minimum k for graph G which has a vertex irregular reflexive k-labeling is called the reflexive vertex strength of the graph G, denoted by rvsG. In this paper, we determined the exact value of the reflexive vertex strength of any graph with pendant vertex which is useful to analyse the reflexive vertex strength on sunlet graph, helm graph, subdivided star graph, and broom graph.

CGANT Research Group—the University of Jember
1. Introduction

We consider a simple and finite graph G=V,E with vertex set VG and edge set EG. We motivate the readers to refer Chartrand et al. , for detailed definition of the graph. A topic in graph theory which has grown fast is the labeling of graphs. The concept of graph labeling, firstly, was introduced by Wallis in . He defined a labeling of G is a mapping that carries a set of graph elements into a set of integers called labels. By this definition, we can have a vertex label, edge label, or both of them. Baca et al.  introduced the total labeling, and they defined the vertex weight as the sum of all incident edge labels along with the label of the vertices. Many types of labeling have been studied by researchers, namely, graceful labeling, magic labeling, antimagic labeling, irregular labeling, and irregular reflexive labeling.

Furthermore, labeling known as a vertex irregular total k-labeling and total vertex irregularity strength of graph is the minimum k for which the graph has a vertex irregular total k-labeling. The bounds for the total vertex irregularity strength are given in . In , Tanna et al. identified the concept of vertex irregular reflexive labeling of graphs. In this paper, we continue to study a vertex irregular reflexive labeling as there are still many open problems. By irregular reflexive labeling, we mean a labeling of graph which the vertex labels are assigned by even numbers from 0,2,,2k and the edge labels are assigned by 1,2,3,,k, where k is positive integer. The weight of each vertex, under a total labeling, is determined by summing the incident edge labels and the label of the vertex itself.

A k-labeling assigns numbers 1,2,,k to the elements of graph. Let k be a natural number, a function f:VGEG1,2,3,,k is called total k-irregular labeling. Hinding et al.  defined that a total labeling ϕ:VGEG1,2,3,,k is called vertex irregular total k-labeling of graph G if the vertex weight wtϕx=ϕx+ΣxyEGϕxy is distinct for every two different vertices, wtϕxwtϕy for x,yVG,xy. The minimum k for which graph G has a vertex irregular total k-labeling is called total vertex irregularity strength, denoted by tvsG.

The concept of vertex irregular total k-labeling extends to a vertex irregular reflexive total k-labeling. The definition of total k-labeling is a function fe from the edge set to the first natural number ke and a function fv from the vertex set to the nonnegative even number up to 2kv, where k=maxke,2kv. A vertex irregular reflexivek-labeling of the graph G is the total k-labeling, for every two different vertices x and x of G, wtxwtx, where wtx=fvx+ΣxyEGfexy. The minimum k for graph G which has a vertex irregular reflexive k-labeling is called the reflexive vertex strength of the graph G, denoted by rvsG.

Some results related to vertex irregular reflexive labeling have been studied by several researchers. Tanna et al.  have studied the vertex irregular reflexive of prism and wheel graphs, Ahmad and Bac̆a  have studied the total vertex irregularity strength for two families of graphs, namely, Jahangir graphs and circulant graphs, and Agustin et al.  also study the concept of vertex irregular reflexive labeling of cycle graph and generalized friendship. Another results of irregular labeling can be seen on . In this paper, we have found the lower bound of vertex irregular reflexive strength of any graph G and determined the vertex irregular reflexive strength of graphs with pendant vertex. Our results are started by showing one lemma and theorem which describe a general construction of the existence of vertex irregular reflexive k-labeling of graph with pendant vertex.

2. Result and Discussion

The following lemma and theorem will be used as a base construction of analysing the reflexive vertex strength of any graph with pendant vertex, namely, sunlet graph, helm graph, subdivided star graph, and broom graph.

Lemma 1.

For any graph G of order p, the minimum degree δ, and the maximum degree Δ,(1)rvsGp+δ1Δ+1.

Proof. Let G be a graph of order p, the minimum degree δ, and the maximum degree Δ. The total k-labeling which labeling f defined fe:EG1,2,,ke and fv:VG0,2,,2kv such that fx=fvx if xVG and fx=fex if xEG, where k= maxke,2kv. The total k-labeling f is called a vertex irregular reflexive k-labeling of the graph G if every two different vertices x and x, and it holds wtxwtx, where wtx=fvx+ΣxyEGfexy. Furthermore, since we require k-minimum for the graph G which has a vertex irregular reflexive labeling, the set of a vertex weight should be consecutive, otherwise it will not give a minimum rvs. Thus, the set of a vertex weight is Wtx=δ,δ+1,δ+2,,12kv+Δke. Since the minimum k=maxke,2kv is the reflexive vertex strength, the maximum possible vertex weight of graph G is at most k1+Δ. It implies 2kv+Δkeδ+p1.1k+Δkδ+p1kδ+p1/Δ+1. Since rvsG should be integer and we need a sharpest lower bound, it implies(2)rvsGδ+p1Δ+1.

It completes the proof.

Theorem 1.

Let G be a graph of order n and contains l pendant vertex. If lnl, then(3)rvsG=l2+1,for l even and l2 odd,l2,otherwise.

Proof.

Given that a graph G of order n is with l pendant vertices. The labeling of graph G is with respect to two components, namely, the pendant vertices and otherwise vertices. Thus, we will split our proof into two cases.

Case 1.

Let Vl be a set of pendant vertices and the number of Vl is l. A pendant vertex consists of two elements, i.e., a vertex and an edge. The vertex weight on each pendant vertex must be different. Suppose we choose those vertex weights are 1,2,,l. Those vertex weights are obtained by summing the vertex and edge labels. To prove the above rvsG, let us suppose the maximum vertex weight of l. Let(4)l2=l+12,for l odd,l2,for l even.

Define an injection f as the labels. Since the weight l is contributed by one vertex and one edge labels, it will give four possibilities.

If l+1/2 is odd number, then fv=l+1/21 and fe=l+1/2, such that the vertex weight is l+1/21+l+1/2=l+11=l

If l+1/2 is even number, then fv=l+1/2 and fe=l+1/2, such that the vertex weight is l+1/2+l+1/2=l+1

If l/2 is odd number, then fv=l/21 and fe=l/2, such that the vertex weight is l/21+l/2=l1

If l/2 is even number, then fv=l/2 and fe=l/2, such that the vertex weight is l/2+l/2=l

The vertex weight of point 1,2, and (4) are, respectively, l and l+1. It will give all weights are different, whereas, point (3) has a vertex weight of l1. Since the number of pendants is l, we will have at least two vertices which have the same weight. Therefore, for l is even and l/2 is odd, we need to add 1 for the largest vertex or edge labels. Thus, we will have a different weight for every pendant. Thus, the labels of vertex and edge of the pendant are the following.

From Table 1, it is easy to see that all vertex weights are different.

Labeling of vertex and edge on pendant vertices.

viv1v2v3vkvk+1vk+2vk+3vk+4vl1vl
fv00002244kk
fe123kk1kk1kk1k
wvi123kk+1k+2k+3k+42k12k
(5)k=l2+1,for l even and l2 odd,l2,otherwise.Case 2.

The vertex set apart from pendant vertices VGVl must have a degree of at least two. The cardinality of VGVl is less than or equal to the cardinality of Vl. It implies that the vertex or edge labels of pendant vertices can be re-used on labels of VGVl. Thus, the vertex weight of VGVl will be different with the vertices of Vl since it has more combination, namely, 2k+1,2k+2,,n.

Based on Case 1 and Case 2, the reflexive vertex strength of graph G is(6)rvsG=l2+1,for l even and l2 odd,l2,otherwise.

It concludes the proof.

Corollary 1.

Let Sn be a sunlet graph, and for every n3,(7)rvsSn=n2+1,if n even and n2 odd,n2,otherwise.

Proof.

Moreover, to determine the label of vertices VSn=ui,vi;1in and edge set ESn=uivi,1inuiui+1,1in1u1un, we will use (Algorithm 1)(8)k=n2+1,if n even and n2 odd,n2,otherwise.

It concludes the proof.

For an illustration, see Figure 1.

<bold>Algorithm 1:</bold> The vertices’ and edges’ labels.

Define vVG, eEG and injecton f for labeling of the graph elements

Assign the labels of vertices and edges of pendants vi according to Theorem 1.

Observe that the vertex weight on each pendant will be 1,2,,k,k+1,,n=2k or 1,2,,k,k+1,,n=2k1.

The vertex weights of point (4),(7) are l, but 5, 6 are respectively is l+1 and l1. Since the number of pendant vertices is l, it will exist two type of vertices which have the same weights. Do the following.

When l is even and l/2 is odd, add the label of each vertex or edge by 1.

Apart from pendants, assign the label of vertices ui as the label of pendant vertices vi, but the labels of edges uiui+1,1in1 with k, thus the vertex weights are 2k+1,2k+2,,3k,3k+1,4k or 2k+1,2k+2,,3k,3k+1,4k1.

Observe, all pendant vertices and otherwise show a different vertex weight.

STOP.

The illustration of labeling on S7 and S12.

Theorem 2.

Let Hnbe a helm graph, and for everyn3,(9)rvsHn=n2+1,if n even and n2 odd,n2,otherwise.

Proof.

Let Hn be a helm graph with vertex set VHn=A,ui,vi;1in, VHn=2n+1 and edge set EHn=Aui,uivi,1inuiui+1,1in1u1un,EHn=3n. Helm graph has n pendant vertices and one central vertex of degree n. Since the central vertex has degree of much greater than the other vertices, it must have a different vertex weight than the others. Based on Theorem 1, we have the following lower bound:(10)rvsHnn2+1,if n even and n2 odd,n2,otherwise.

Furthermore, we will show the upper bound of vertex irregular reflexive k-labeling by defining the injection f and g in the following:(11)fA=0,fui=fvi,fvi=0,if 1ik,2ik2,if k+1in,gAui=guiui+1=gu1un=k,guivi=i,if 1ik,k1,if k+1in,i odd for k even and i even for k odd,k,if k+1in,i even for k even and i odd for k odd,where(12)k=n2+1,if n even and n2 odd,n2,otherwise.

Based on the above injection, the overall vertex weight sets are(13)wvi=i,wvi=i+3k,wA=nk.

It is easy to see that the above elements of set are all different. It concludes the proof.

Theorem 3.

SSn be a subdivided star graph, and for every n3,(14)rvsSSn=2n3.

Proof.

Let SSn be a subdivided star graph with vertex set VSSn=A,xi,yi;1in, VSSn=2n+1 and edge set ESSn=Ayi,xiyi,1in,EHn=2n. The maximum degree of SSn is n. The graph SSn has one central vertex of degree n. Since the central vertex has degree of much greater than the other vertices, it must have a different vertex weight than the others. Based on Lemma 1, we have the following lower bound:(15)rvsGp+δ1Δ+1=2n+112+1=2n3.

For the illustration of the vertex irregular reflexive, k-labeling of SS3 and SS4 can be depicted in Figure 2.

Furthermore, we will show the upper bound of vertex irregular reflexive k-labeling by defining the injection f and g. For n5, we have the following:(16)fA=0,fxi=0,if 1ik,2ik2,if k+1in.

For n3,4mod6, we have the following function for yi and Ayi:(17)fyi=2nk2,if 1ik,2nk2+2ik2,if k+1in1,2nk2+2n1k2,if i=n,gAyi=k1,if 1in2,k,if n1in.

For otherwise n, we have(18)fyi=2nk2,if 1ik,2nk2+2ik2,if k+1ingAyi=k1,if 1in for n5mod6,k,if 1in for otherwise,gxiyi=i,if 1ik,k1,if k+1in,i odd for k evenandi even for k odd,k,if k+1in,i even for k evenandi odd for k odd,where(19)k=2n3.

Based on the above injection, the overall vertex weight sets of the subdivided star SSn are(20)wxi=i,wyi=n+i,if 1in2 for n3,4mod6,2n1,if i=n for n3,4mod6,2n,if i=n1 for n3,4mod6,n+i,if 1in for otherwise,wA=nk1+2,if n3,4mod6,nk1,if n5mod6,nk,otherwise.

It is easy to see that the above elements of the set are all different. It concludes the proof.

The illustration of labeling on SS3 and.SS4.

Theorem 4.

Let Brn,m be a broom graph, and for every n2,n+1m,(21)rvsBrn,m=n+m3+1,if n+m3mod6,n+m3,otherwise.

Proof.

Let Brn,m be a broom graph with vertex set VBrn,m=A,vi,uj;1in,1jm, VBrn,m=n+m+1, and edge set EBrn,m=Avi,1inAumujuj+1,1jm1,EHn=n+m.

The Broom graph Brn,m has n pendant vertices and one central vertex of degree n. Since the central vertex has a degree much greater than the other vertices, it must have a different vertex weight than the others. Based on Lemma 1, we have the following lower bound:(22)rvsBrn,mp+δ1Δ+1=n+m3,for n+m3mod6, n+m=3k and k is odd. Since the vertices uj apart from vertex A have degree of at most 2, the labels of uj are n+m/31, and the label of edges, which are incident to uj, are n+m/3. Thus, the vertex weight uj is 3n+m/31=n+m1. Furthermore, since the number of vertices of Broom graph is n+m, there must be at least two vertices with the same vertex weight. Thus, we need to add 1 on the sharpest lower bound:(23)rvsBrn,mn+m3+1,if n+m3mod6.

Furthermore, we will show that k is an upper bound of the reflexive vertex strength of Broom graph Brn,m. Let(24)k=n+m3+1,if n+m3mod6,n+m3,otherwise.

Define an injection f and g of the vertex irregular reflexive k-labeling of Broom graph rvsBrn,m as follows:(25)fA=k1,if k odd,k,if k even,fvi=0,if 1ik,2ik2,if k+1in+1,fu1=fvn+1,gAvi=i,if 1ik,k1,if k+1in+1,i odd for k evenandi even for k odd,k,if k+1in+1,i even for k evenandi odd for k odd,gu1u2=gAvn+1.

Based on the above injection, the overall vertex weight sets of Brn,m for vi;1in+1 is(26)wvi=i,1in+1.

Moreover, to determine label of vertices VG=uj,2jm and EG=Aumujuj+1,1jm1, we will use Algorithm 2.

It is easy to see that the above elements of set wvi and wuj are all different. It concludes the proof.

<bold>Algorithm 2:</bold> The vertices’ and edges’ labels.

Given that the vertex weight VG=uj,2jm by wuj=n+j,2jm.

Assign the labels of vertices uj by fuj=fUj1 and assign the labels of edges which are incident to uj by 1,2,3,,k such that it meets with given vertex weight.

When on point (ii), the label of edges is more than k, relabel the vertices with fuj=fuj1+2 as well as relabel the edges which are incident to uj by 1,2,3,,k such that it meets with given vertex weight wuj=n+j,2jm.

STOP.

3. Concluding Remark

In this paper, we have studied the construction of the reflexive vertex k-labeling of any graph with pendant vertex. We have determined a sharp lower bound of the reflexive vertex strength of any graph G in Lemma 1, as well as obtained the exact value the reflexive vertex strength of any graph G in Theorem 1. By this lemma and theorem, we finally determined the reflexive vertex k-labeling of some families of graph with a pendant vertex. However, we need to find an upper bound of the reflexive vertex strength of any graph and study the reflexive vertex k-labeling of other families of graph or some graph operations. Therefore, we propose the following open problems:

Determine an upper bound of reflexive vertex strength of any graph to find the gap between lower bound and upper bound, and continue to determine the exact values for reflexive vertex strength of any other special graphs

Determine the construction of the reflexive vertex k-labeling of any regular graph, planar graph, or some graph operations

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors gratefully acknowledge from the support of CGANT Research Group—the University of Jember, Indonesia of year 2020.

ChartrandG.LesniakL.ZhangP.Graphs and Digraphs, Sixth2016New York, NY, USATaylor and Francis GroupWallisW. D.Magic Graphs2001Boston, MA, USABirkhauserBac̆aM.JendrolS.MillerM.RyanD. J.On irregular total labellingsDiscrete Mathematics200730713781388TannaD.RyanJ.Semaničová-FeňovčíkováA.BačaM.Vertex irregular reflexive labeling of prisms and wheelsAKCE International Journal of Graphs and Combinatorics2018In pressHindingN.FirmayasariD.BasirH.BačaM.Semaničová-FeňovčíkováA.On irregularity strength of diamond networkAKCE International Journal of Graphs and Combinatorics201815329129710.1016/j.akcej.2017.10.0032-s2.0-85032329407AhmadA.Bac̆aM.On vertex irregular total labelingsArs Combinatoria2013112129139AgustinI. H.UtoyoM. I.DafikMohanapriyaN.On Vertex irregular reflexive labeling of cycle and friendship graphASCM Published By Jangjeon Mathematical Society20202020115HaryantiA.IndriatiD.MartiniDan T. S.On the total vertex irregularity strength of firecracker graphJournal of Physics: Conference Series201912111910.1088/1742-6596/1211/1/0120112-s2.0-85066880717NurdinE. TGaosD. N. N.BaskoroE. T.SalmanA. N. M.GaosN. N.On the total vertex irregularity strength of treesDiscrete Mathematics2010310213043304810.1016/j.disc.2010.06.0412-s2.0-79952984270RamdaniR.SalmanA. N. M.AssiyatunA.On the total irregularity strength of regular graphsJournal of Mathematical and Fundamental Sciences201547328129510.5614/j.math.fund.sci.2015.47.3.62-s2.0-84953732527SlaminD.WinnonaW.Total vertex irregularity strength of the disjoint union of sun graphsThe Electronic Journal of Combinatorics2011201219SusilawatiE. T. B.SimanjuntakD.BaskoroE. T.SimanjuntakR.TotDl vertex-irregularity labelings for subdivision of several classes of treesProcedia Computer Science20157411211710.1016/j.procs.2015.12.0852-s2.0-84964017516SusilawatiS.BaskoroE. T.BaskoroE. T.SimanjuntakR.Total vertex irregularity strength of trees with maximum degree fiveElectronic Journal of Graph Theory and Applications20186225025710.5614/ejgta.2018.6.2.52-s2.0-85057947575AgustinI. H.DafikMarsidiAlbirriE. R.On the total H-irregularity strength of graphs: a new notionJournal of Physics: Conference Series201785501200410.1088/1742-6596/855/1/0120042-s2.0-85023644895NisviasariR.AgustinI. H.The total H-irregularity strength of triangular ladder and grid graphsJournal of Physics: Conference Series2019121101200510.1088/1742-6596/1211/1/0120052-s2.0-85066912405