INFOKARA RESEARCH ISSN NO: 1021-9056

SUM OF AN ABSOLUTE DIFFERENCES OF CUBIC AND SQUARE DIFFERENCE LABELING FOR PATH RELATED GRAPHS

P.Shalini1, S.Sri Harini2, Dr.D.Paul Dhayabaran3

1Asst Professor, Cauvery College For Women, Tiruchirapalli-18, India.

2PG Student, Cauvery College For Women, Tiruchirapalli-18, India.

3Principal, Bishop Heber College, Tiruchirapalli-17, India.

ABSTRACT A graph G = (V,E) with p vertices and q edges is said to be sum of an absolute differences of cubic and square difference labeling, if there exists a bijection f : E(G) → even numbers given by f(uv) = |[f(u)]3 − [f(v)]3 | + |[f(u)]2 − [f(v)]2 | for every uv ϵ E(G) are all distinct even numbers. In this paper, we introduce the new concept, sum of an absolute difference of cubic and square difference labeling of a path related graphs.

KEYWORDS

Graph labeling, Cubic Difference Labeling, Square Difference Labeling.

1. INTRODUCTION

The field of plays a very important role in different fields of Science and Engineering. One of the most important areas in mathematics is . In graph theory, one of the main concepts is . A graph can be labeled or unlabeled. Labeled graphs are used to identification. Labeling can be used not only to identify vertices or edges, but also to signify some additional properties depending on the particular labeling. Graph labeling is an assignment of , to its vertices or edges subject to some certain conditions.

Volume 9 Issue 2 2020 58 http://infokara.com/ INFOKARA RESEARCH ISSN NO: 1021-9056

All graphs in this paper are finite and undirected. The symbols V (G) and E (G) denotes the and edge set of a graph G. Some basic definitions and notations are taken from Gary Chatrand [5]. A dynamic survey on graph labeling is regularly updated by Gallian [4] and it is published by Electronic journal of combinatorics. Some basic concepts are taken from Frank Harary [3]. Shalini and Paul Dhayabaran [10] introduced the concept an absolute difference of cubic and square difference labeling. In this paper, the new concept for sum of an absolute difference of cubic and square difference in graph labeling has been introduced. New definition and formula for sum of an absolute difference of cubic and square difference in graph labeling have been established. In addition, few graphs are characterized for the sum of an absolute difference of cubic and square difference graphs

DEFINITION 1.1

Let G = (V(G),E(G)) be a graph. A graph G is said to be a sum of an absolute difference of cubic and square difference labeling, if there exists a bijection f: V(G) →{1,2,3,……,P} such that the induced function f *: E(G) → even numbers is given by,

f*(uv) = | [f(u)]3 − [f(v)]3 | + | [f(u)]2 − [f(v)]2 |

i.e) f *(uv) = | [f ꞌ(uv) + f ꞌꞌ(uv) |

Where , f ꞌ(uv) = | [f(u)]3 − [f(v)]3 |

f ꞌꞌ(uv) = | [f(u)]2 −[f(v)]2 |

DEFINITION 1.2

A graph in which every edge associates a distinct value with even numbers is said to be a sum of an absolute difference of cubic and square difference labeling.

Volume 9 Issue 2 2020 59 http://infokara.com/ INFOKARA RESEARCH ISSN NO: 1021-9056

DEFINITION 1.3

A graph in which every edge associates a distinct value with multiples of 2 is called the sum of the differences of the cubes of the vertices and the differences of the squares of the vertices.

DEFINITION 1.4

The ladder graph Ln is a planar undirected graph, and the

ladder graph with 2n vertices and 3n – 2 edges.The ladder graph Ln can be obtained as the Cartesian product of two path graphs, one of which has only one edge.

i.e.) Ln,1 = Pn × P2.

2. SUM OF AN ABSOLUTE DIFFERENCE OF CUBIC AND SQUARE DIFFERENCE LABELING

THEOREM 2.1

The Tn is a sum of an absolute differences of cubic and square difference graphs n ≥ 2. Proof

Let G be a graph of tree Tn.

Let {v0,v1,v2,...... ,vn,vn+1} be the vertices of Tn and

{e1, e2, e3, . . . , en, en+1} be the edges of Tn which are denoted as in the Figure 2.1.,

Volume 9 Issue 2 2020 60 http://infokara.com/ INFOKARA RESEARCH ISSN NO: 1021-9056

e en-3 4 . . . .

v4 vn-3 e3 en-2

v e2 e 3 n-1 vn-2

e1 en v2 v v0 n-1

vn v1 en+1

vn+1 Fig 2.1 : Tree Tn with ordinary labeling

The tree Tn consists of n + 2 vertices and n + 1 edges.

The vertices of Tn are labeled as given below. Define f : V(G)  {1, 2, 3, . . . , n+2} by

f(v0) = 1

f(vi) = i + 1 ; 1 ≤ i ≤ n + 1 Then the induced edge labels are: 2 f(ei) = 3(i+1) – 2 − (i − 1) ; 1 ≤ i ≤ n + 1 The edges of the tree graph receive distinct even numbers.

Hence, the tree Tn are the sum of an absolute differences of cubic and square difference graphs n ≥ 2. EXAMPLE 2.1

Fig 2.2 Tree T7

Volume 9 Issue 2 2020 61 http://infokara.com/ INFOKARA RESEARCH ISSN NO: 1021-9056

THEOREM 2.2

The star K1,n is a sum of an absolute differences of cubic and square difference graphs n ≥ 2. Proof

Let G be a graph of star K1,n.

Let v0 be the centre vertex and {v1, v2, v3, . . ., vn} be the pendant

vertices of K1,n and {e1, e2, e3, . . ., en} be the edges of K1,n which are denoted as in the Figure 2.3.,

v1 vn v2 vn-1 v3

vn-2 e 1 e 2 n v e 4 e 3 n e -1 v e 5 n -2 e 4 e 5 v6

e6 v0 v7 e7

e 8

0 1 e e9

v8

v10 v9

Fig 2.3 : Star K1,n with ordinary labeling

The star K1,n consists of n + 1 vertices and n edges.

The vertices of K1,n are labeled as given below. Define f : V(G)  {1, 2, 3, . . . , n+1} by

f(v0) = 1

f(vi) = i + 1 ; 1 ≤ i ≤ n Then the induced edge labels are: 3 2 f(ei) =2 − (i+1) − (i+1) ; 1 ≤ i ≤ n The edges of the star graph receive distinct even numbers.

Hence, the star K1,n are the sum of an absolute differences of cubic and square difference graphs n ≥ 2

Volume 9 Issue 2 2020 62 http://infokara.com/ INFOKARA RESEARCH ISSN NO: 1021-9056

EXAMPLE 2.2

Fig 2.4 Star K 1,8 THEOREM 2.3

The path Pn is sum of an absolute differences of cubic and square difference graphs for n ≥ 2. Proof

Let G be a graph of Path Pn.

Let {v1, v2, v3, . . ., vn} be the vertices of Pn and{e1, e2, e3,..., en-1 }

be the edges of Pn which are denoted as in the Figure 2.5.,

e1 e2 ...... en-3 en-2 en-1

v1 v2 v3 vn-3 vn-2 vn-1 vn

Fig 2.5 : Path Pn with ordinary labeling

The path Pn consists of n vertices and n-1 edges.

The vertices of Pn are labeled as given below. Define f : V(G)  {1, 2, 3, . . . , n} by

f(vi) = i ; 1 ≤ i ≤ n Then the induced edge labels are: 2 f(ei) = 3(i+1) − 2 − (i − 1) ; 1 ≤ i ≤ n−1 3 2 f(en ) = f(vnv1) = 2 − (n+1) − (n+1)

Volume 9 Issue 2 2020 63 http://infokara.com/ INFOKARA RESEARCH ISSN NO: 1021-9056

The edges of the receive distinct even numbers.

Hence, the path Pn(n ≥ 2) are sum of an absolute differences of cubic and square difference graphs. EXAMPLE 2.3

Fig 2.6 Path P8

THEOREM 2.4

The ladder Ln is a sum of an absolute difference of cubic and square difference graphs n ≥ 3.

Proof

Let G be a graph of Ladder Ln .

v , v , v , . . . . , v {e , e , Let {v1,v2, ……,vn, 1 2 3 n } be the vertices of Ln and 1 2

e3, . . . , en − 1, e1 , e2 , e3 , . …, en-1 , a1,a2,……..an} be the edges of Ln which are denoted as in the figure 2.7.,

Volume 9 Issue 2 2020 64 http://infokara.com/ INFOKARA RESEARCH ISSN NO: 1021-9056

Fig 2.7 Ladder Ln with ordinary labeling,

The ladder Ln consists of 2n vertices and 3n − 2 edges.

The vertices of Ln are labeled as given below, Define, f :V(G)→{1,2,…….,2n} by

f(vi) = i ; 1 ≤ i ≤ n

f(viꞌ) = n+1 ; 1 ≤ i ≤ n Then the induced edge labels are:

2 f(ei) = 3(i+1) – 2 – (i−1) ; 1 ≤ i ≤ n−1

3 3 2 2 f(eiꞌ) = (i − (i+1) ) + (i − (i+1) ) ; 1 ≤ i ≤ n−1

3 2 3 2 f(ai) = [− (n+i) − (n+i) ] + [i +i ] ; 1 ≤ i ≤ n

The edges of the ladder graph receive distinct even numbers.

Hence, the ladder Ln are the sum of an absolute difference of cubic and square difference graphs for n ≥ 3.

Volume 9 Issue 2 2020 65 http://infokara.com/ INFOKARA RESEARCH ISSN NO: 1021-9056

EXAMPLE 2.4

Fig 2.8 Ladder L8

CONCLUSION

In this paper, we have discussed a sum of an absolute difference of the cubic and squared difference labeling. We investigated that some families of graph receive weight as a multiples of 2.We conclude that tree, star, path and ladder graphs are sum of an absolute difference of cubic and square difference labeling for path related graphs.

REFERENCES

[1] Ajitha, V., Arumugam, S and Gelmina, K.A. On Square Sum Graphs, AICCEE3 Graphs, Combin,6(2006)1-10

[2] Bloom,G.S., Hsu, D.F., Graceful Directed Graphs, SIAMJ, Alg, Discrete Math., 6(1985), 519-536,

[3] Frank Harrary, Graph Theory, Narosa Publishing House, 2001.

[4] Gallian, J.A, A Dynamic Survey of Graph Labeling, The Electronics Journal of combinatorics, 17(2010)# DS6

[5] Gary Chartrand, Ping Zhang, Introduction to Graph Theory, McGraw-Hill International Edition.

[6] Harary,F., Graph Theory, New Delhi: Narosa Publishing House, 2001.

Volume 9 Issue 2 2020 66 http://infokara.com/ INFOKARA RESEARCH ISSN NO: 1021-9056

[7] Hedge, S.M., Labeled Graphs and Digraphs: Theory and Application.

[8] Lo, S., On Edge- of Graphs, Conger Number, 50 (1985), 231-241.

[9] Rosa, A., On Certain Valuation of the Vertex of a Graph, Theory of Graphs.

[10] Shalini, P., Paul Dhayabaran, D., An Absolute Difference of Cubic and Square Difference Labeling, International Journal of Advanced Scientific and Technical Research, ISSN 2249-9954

[11] Shiama, J. Differences of Squared Labeling, International Journal of Mathematical Archive, 316(2012) 2369-2374.

[12] Weisstein, Eric W. “Ladder Graph”,Mathworld.

Volume 9 Issue 2 2020 67 http://infokara.com/