ON SUPER HERONIAN MEAN LABELING OF GRAPHS

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1 ON SUPER HERONIAN MEAN LABELING OF GRAPHS S.S.Sandhya Department of Mathematics, Sree Ayyappa College for Women, Chunkankadai- 00,Tamilnadu,India. E.Ebin Raja Merly Department of Mathematics, Nesamony Memorial Christian College, Marthandam-,Tamilnadu,India G.D.Jemi Department of Mathematics, Narayanaguru College of Engineering, Manjalumoodu-,Tamilnadu,India. ABSTRACT Here we focus on Super Heronian Mean Labeling of graphs.we shall render brief summary of definitions and other information which are necessary and useful for the present investication. Key words: Graph, Super Heronian mean graph, L n K,, T L n K, T n K, D(T n ) K. Introduction : We consider simple,finite,undirected and connected graph G=(V,E).In this paper L n denotes Ladder with n vertices.for all other general expressions and symbols we follow Harary.First we will provide some definitions useful for the present work. Definition:. Let f :V(G) {,,------,p+q} be an injective function. For a vertex labeling f the induced edge labeling f*(e=uv) is defined by,

2 f*(e)= ( ) ( ) ( ) ( ) [OR] ( ) ( ) ( ) ( ) Then f is called a Super Heronian Mean Labeling if {f(v(g)} U {f(e): e ϵ E(G)={,,...,p+q}}. A graph which admits Super Heronian Mean Labeling is called Super Heronian Mean Graph. Definition:. If the vertices of the graph are assigned values subject to certain conditions is known as Graph Labeling Definition:. The product P xp n is called a Ladder and it is denoted by L n. Definition:. A triangular snake T n is obtained from a path u u... u n joining u i to u i+ and to a new vertex v i for i n- ie, every edge of a path is replaced by a triangle C. Theorem :. Ladders are Super Heronian Mean Graphs. Theorem :. Triangular snake T n are Super Heronian Mean Graphs..Main Results: Theorem:. L n K, is a Super Heronian mean labeling. Let L n be a Ladder.Let w i and x i be the pendent vertex adjacent to u i and y i and z i be the pendant vertex adjacent to u i. Define a function, f: V( L n K, ) {,,...,p+q} by, f(u i )=i- ; i n f(v i )=i- ; i n f(w i )=i- ; i n f(x i )=i- ; i n f(y i )=i- ; i n f(z i )=i- ; i n f(u i u i+ )=i+ ; i n- f(v i v i+ )=i- ; i n- f(u i w i )=i- ; i n f(u i x i )=i- ; i n f(v i u i )=i- ; i n f(v i y i )=i-0 ; i n f(v i z i )=i- ; i n Hence L n K, is a Super Heronian mean labeling.

3 Example:. A Super Heronian mean labeling of L K, is given below Figure : Theorem:. Triangular Ladder TL n is a Super Heronian mean graph. Let TL n be a Triangular Ladder. Let u u... u n and v v... v n be two paths of length n in the graph TL n. Join u i v i+, i n-. Define a function f: V(T L n ) {,,...,p+q} by, f(u i )=i- ; i n f(v i )=i- ; i n f(u i u i+ )=i ; i n- f(u i v i )=i- ; i n f(v i v i+ )=i- ; i n- f(v i u i+ )=i- ; i n- Obviously f is a Super Heronian mean labeling and TL n is a Super Heronian mean graphs.

4 Example:. A Super Heronian mean labeling of TL is dispayed below. 0 Figure : Theorem:. TL n Let TL n be a Triangular Ladder.Let u u... u n and v v... v n be two paths of length n in the graph TL n K. Join u i v i+ ; i n-.let w i, x i be the pendant vertices.join v i w i and u i x i. Define a function f:v(tl n K ) {,,...,p+q} by, f(u i )=0i- ; i n f(v i )=0i- ; i n f(w )= ; f(w i )=0i- ; i n f(x i )=0i- ; i n f(u i v i )=0i- ; i n f(u i u i+ )=0i+ ; i n- f(v i v i+ )=0i- ; i n- f(v i u i+ )=0i ; i n- f(v w )= ; f(v i w i )=0i- ; i n f(u i x i )=0i- ; i n Obviously f is a Super Heronian mean labeling, and TL n 0

5 Example:. A Super Heronian mean labeling of TL K is shown below Figure : Theorem:. T n Let T n be a Triangular snake.let u i,v i be the vertices of a Triangular snake. Join u i v i and u i+ v i.let w i, x i be the pendant vertices.join u i w i and v i x i, i n ; i n- Define a function f:v(t n K ) {,,...,p+q} by, f(u i )=i- ; i n f(v i )=i- ; i n- f(w i )=i- ; i n f(x i )=i- ; i n- f(u i u i+ )=i- ; i n- f(u i v i )=i- ; i n- f(u i+ v i )=i ; i n- f(u i w i )=i- ; i n f(v i x i )=i- ; i n- Obviously f is a Super Heronian mean labeling, and T n

6 Example:. A Super Heronian mean labeling of T K is given below. x w v 0 u Theorem:. Figure : 0 D(T n ) Let D(T n ) be a Double Triangular snake.let u i, v i, w i be the vertices of Double Triangular snake. Join u i v i, u i+ v i, u i w i, u i+ w i.let x i, y i and s i, t i be the pendant vertices.join u i x i, v i y i and u i s i, w i t i i n ; i n-. Define a function f: V(D(T n K ) {,...,p+q} by, f(u i )=i- ; i n, f(v i )=i- ; i n- f(w i )=i- ; i n- f(x i )=i- ; i n f(y i )=i- ; i n- f(s i )=i- ; i n f(t i )=i- ; i n- f(u i u i+ )=i- ; i n- f(u i v i )=i-0 ; i n- f(v i u i+ )=i- ; i n- f(u i w i )=i- ; i n- f(u i+ w i )=i ; i n- f(u i x i )=i- ; i n f(u i s i )=i- ; i n

7 f(v i y i )=i- ; i n- f(w i t i )=i- ; i n- Obviously f is a Super Heronian mean labeling, and D(T n ) Example:.0 A Super Heronian mean labeling of D(T ) K is displayed below 0 x v 0 0 s w 0 x Figure : References: [].J.A.Gallian,A Dynamic Survey of Graph labelling, The Electronic journal of Combinatorics(). []. F.Harary(),Graph Theory,NarosaPublishing House Reading,New Delhi. []. S.Somasundaram and R.Ponraj, Mean Labeling of Graphs,National Academy of Science Letters vol.,p.0-. [].S.Somasundaram,R.Ponraj and S.S.Sandhya, Harmonic Mean Labeling of Graphs communicated to Journal of Combinatorial Mathematics and Combinatorial Combuting. [].C.Jeyasekaran, S.S.Sandhya and C. David raj, Some Results on Super Harmonic Mean Graphs,International Journal of Mathematics trend and Tecnology,vol.()(),-. []. S.Somasundaram,R.Ponraj and P.Vidhyarani, Geometric Mean Labeling of Graphs Bulletin of Pure and Applied Sciences 0E(z)()p.-0. []. S.S.Sandhya, E.Ebin Raja Merly and G.D.Jemi, Super Heronian Mean Labeling of Graphs communicated to International Journal of Mathematical Forum.