SUPER ROOT SQUARE MEAN LABELING OF SOME NEW GRAPHS

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1 SUPER ROOT SQUARE MEAN LABELING OF SOME NEW GRAPHS 1 S.S.Sandhya S.Somasundaram and 3 S.Anusa 1.Department of Mathematics,Sree Ayyappa College for women,chunkankadai:69003,.department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli:6701, 3.Department of Mathematics, Arunachala College of Engineering for Women,Vellichanthai-6903, ABSTRACT Let G be a p, q graph and f: V(G) 1,,3,, p + q be an injective function. For each edge = uv, let f e = uv = f(u) +f(v) or f(u) +f(v), then f is called a super root square mean labeling if f V f e : e E G = 1,,, p + q. A graph that admits a super root square mean labeling is called as super root square mean graph. In this paper we prove that P n K 1,, P n K 1,3, TL n, T n K 1, Q n K 1 are super root square mean graphs. Key Words: 1. Introduction Root Square mean graph, Super Root Square mean graph, Path, Triangular snake, Quadrilateral snake. All graphs in this paper are finite, simple and undirected graph G = V, E with p vertices and q edges. For all detailed survey of graph labeling we refer to Gallian [1]. For all other standard terminology and notations we follow Harary []. The concept of Root Square mean labeling was introduced by S.S.Sandhya, S.Somasundaram and S.Anusa in [3] and proved many results in [4,5,6,7,8,9]. In this paper we proved that P n K 1,, P n K 1,3, TL n, T n K 1, Q n K 1 are super root square mean graphs. The following definitions and theorems are useful for the present study. Definition 1.1: Let G be a p, q graph and f: V(G) 1,,3,, p + q be an injective function. For each edge = uv, let f e = uv = f(u) +f(v) or f e : e E G = 1,,, p + q. f(u) +f(v), then f is called a super root square mean labeling if f V Page 16

2 Definition1.: A Triangular Ladder is a graph obtained from L n by adding the edges u i v i+1, 1 i n 1, where u i and v i, 1 i n are the vertices of L n such that u 1 u u n and v 1 v v n are two paths of length n in the graph L n. Definition1.3: A Triangular snake T n is obtained from a path u 1 u u n by joining u i and u i+1 to a new vertex v i for 1 i n 1. Definition1.4: A Quadrilateral snake Q n is obtained from a path u 1 u u n by joining u i and u i+1 to two new vertices v i and w i 1 i n 1 respectively and then joining v i and w i. Definition1.5: The Corona of two graphs G 1 and G is the graph G = G 1 G formed by taking one copy of G 1 and V(G 1 ) copies of G where the i th vertex of G 1 is adjacent to every vertex in the i th copy of G. Theorem 1.6: Any path P n is a Super Root Square mean graph. Theorem 1.7: Ladder L n is a Super Root Square mean graph. Theorem 1.8: Triangular Snake T n is a Super Root Square mean graph. Theorem 1.9: Quadrilateral Snake Q n is a Super Root Square mean graph..main Results Theorem.1: Triangular Ladder TL n is a Super Root Square Mean graph. Proof: Let u 1, u,, u n and v 1, v,, v n be two paths of length n. Join u i and v i, 1 i n. Join u i and v i+1, 1 i n 1.The resulting graph is TL n. Define a function f: V(TL n ) 1,,, p + q by f u 1 = 1, f u i = 6i 6, i n f v i = 6i 3, 1 i n f u i u i+1 = 6i, 1 i n 1 f v i v i+1 = 6i + 1, 1 i n 1 f u i v i = 6i 4, 1 i n 1 f u i v i+1 = 6i 1, 1 i n 1 Then f V f e : e E G = 1,,, p + q. Hence by definition 1.1,Triangular Ladder TL n is a Super root square mean graph. Page 163

3 Example.: The labeling pattern of TL 5 is shown below. Figure 1 Theorem.3: P n K 1, is a super root square mean graph. Proof: Let u 1, u,, u n be the path P n. Let v i and w i, 1 i n be the vertices of K 1, attached to u i. Define a function f: V(P n K 1, ) 1,,, p + q by f u i = 6i 3, 1 i n f v i = 6i 5, 1 i n f w i = 6i 1, 1 i n f u i u i+1 = 6i, 1 i n 1 f u i v i = 6i 4, 1 i n f u i w i = 6i, 1 i n Then f V f e : e E G = 1,,, p + q. Hence by definition 1.1, P n K 1, is a Super root square mean graph. Example.4: Super root square mean labeling of P 4 K 1, is shown below. Figure Theorem.5: P n K 1,3 is a super root square mean graph. Proof: Let u 1, u,, u n be the path P n. Let v i,w i and s i, 1 i n be the vertices of K 1,3 attached to u i. Define a function f: V(P n K 1,3 ) 1,,, p + q by Page 164

4 f u i = 8i 5, 1 i n f v i = 8i 7, 1 i n f w i = 8i 3, 1 i n f s i = 8i 1, 1 i n f u i u i+1 = 8i, 1 i n 1 f u i v i = 8i 6, 1 i n f u i w i = 8i 4, 1 i n f u i s i = 8i, 1 i n Then f V f e : e E G = 1,,, p + q. Hence by definition 1.1, P n K 1,3 is a Super root square mean graph. Example.6: Super root square mean labeling of P 4 K 1,3 is shown below. Theorem.7: T n K 1 is a Super Root Square Mean graph. Figure 3 Proof: Let u 1, u,, u n be a path of length n. Let v i, 1 i n 1 be the new vertex joined to u i and u i+1.the resulting graph is called T n. Let x i be the vertex which is joined to u i, 1 i n. Let y i be the vertex which is joined to v i, 1 i n 1. The resulting graph is T n K 1. Let G = T n K 1. Define a function f: V G {1,,, p + q} by f u i = 9i 6, 1 i n f v i = 9i 4, 1 i n 1 f x i = 9i 38, 1 i n f y i = 9i, 1 i n 1 f u i u i+1 = 9i 1, 1 i n 1 f u i v i = 9i 5, 1 i n 1 Page 165

5 f v i u i+1 = 9i, 1 i n 1 f u i x i = 9i 7, 1 i n f v i y i = 9i 3, 1 i n 1 Then f V f e : e E G = 1,,, p + q. Hence by definition 1.1, T n K 1 is a Super Root Square Mean graph. Example.8: The Super Root Square Mean labeling of T 4 K 1 is given below. Theorem. 9: Q n K 1 is a Root Square Mean graph. Figure 4 Proof: Let u 1, u,, u n be a path. Let v i and w i be two vertices joined to u i and u i+1 respectively and then join v i and w i, 1 i n 1. The resulting graph is called as quadrilateral snake Q n. Let x i be the new vertex joined to u i, 1 i n. Let y i be the new vertex joined to v i, 1 i n 1. Let z i be the new vertex joined to w i, 1 i n 1. The resulting graph is Q n K 1. Let G = Q n K 1. Define a function f: V G {1,,, p + q} by f u i = 13i 10, 1 i n f v i = 13i 8, 1 i n 1 f w 1 = 11, f w i = 13i 1, i n 1 f x 1 = 1, f x i = 13i 13, i n f y i = 13i 6, 1 i n 1 f z i = 13i 4, 1 i n 1 f u 1 u = 1, f u i u i+1 = 13i, i n 1 f u i x i = 13i 11,1 i n f u i v i = 13i 9,1 i n 1 Page 166

6 f u i+1 w i = 13i + 1,1 i n 1 f v i w i = 13i 5,1 i n 1 f v i y i = 13i 7,1 i n 1 f w i z i = 13i 3,1 i n 1 Then f V f e : e E G = 1,,, p + q. Hence by definition 1.1, Q n K 1 is a Super Root Square Mean graph. Example.10: The labeling pattern of Q 4 K 1 is given below. Reference: Figure 5 [1]Gallian. J.A, 01, A dynamic Survey of graph labeling. The electronic Journal of Combinatories. []Harary.F, 1988, Graph Theory, Narosa Publishing House Reading, New Delhi. [3]Sandhya.S.S, Somasundaram.S, Anusa.S, Root Square Mean Labeling of Graphs International Journal of Contemporary Mathematical Sciences,Vol.9, 014, no.14, [4] Sandhya.S.S, Somasundaram.S, Anusa.S, Some More Results on Root Square Mean Graphs,Journal of Mathematics Research, Vol.7,No.1;015. [5]Sandhya.S.S, Somasundaram.S, Anusa.S, Root Square Mean Labeling of Some New Disconnected Graphs International Journal of Mathematics Trends and Technology, volume 15, number, 014.page no:85-9. [6] Sandhya.S.S, Somasundaram.S, Anusa.S, Root Square Mean Labeling of Subdivision of Some More Graphs International Journal of Mathematics Research, Volume 6, Number 3, [7] Sandhya.S.S, Somasundaram.S, Anusa.S, Some New Results on Root Square Mean Labeling International Journal of Mathematical Archive -5(1), 014, [8] Sandhya.S.S, Somasundaram.S, Anusa.S, Root Square Mean Labeling of Subdivision of Some Graphs Global Journal of Theoretical and Applied Mathematics Sciences, Volume 5, Number 1, (015) pp [9] Sandhya.S.S, Somasundaram.S, Anusa.S, Root Square Mean Labeling of Some More Disconnected Graphs, International Mathematical Forum, Vol.10, 015, no.1, [10] Sandhya.S.S, Somasundaram.S, Anusa.S, Some Results on Super Root Square Mean Labeling Communicated to International Journal of Mathematics and Soft Computing. Page 167