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Sum of an Absolute Differences of Cubic and Square Difference Labeling for Path Related Graphs 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 mathematics plays a very important role in different fields of Science and Engineering. One of the most important areas in mathematics is graph theory. In graph theory, one of the main concepts is graph labeling. 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 integer, 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 vertex set 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 tree 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 path graph 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 . Let {v1,v2, ……,vn, v1 , v2 , v3 , . , vn } be the vertices of Ln and {e1, e2, e3, . , en − 1, e1 , e2 , e3 , . …, en-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.
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