Some Results on Super Heronian Mean. Labeling of Graphs

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1 International Journal of Contemporary Mathematical Sciences Vol., 0, no. 0, - 9 HIKARI Ltd, Some Results on Super Heronian Mean Labeling of Graphs S. S. Sandhya Department of Mathematics Sree Ayyappa College for Women Chunkankadai-9 00, Tamilnadu, India E. Ebin Raja Merly Department of Mathematics Nesamony Memorial Christian College Marthandam-9, Tamilnadu, India G. D. Jemi Department of Mathematics Narayanaguru College of Engineering Manjalumoodu-9, Tamilnadu, India Copyright 0 S. S. Sandhya et al. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We add to some fresh outcomes for Super Heronian Mean Labeling of graphs. It has been found that the graphs obtained by the collection of Triangular snake, Quadrilateral snake also admit Super Heronian Mean Labeling. Keywords: Path, Ladder, Triangular snake, Quadrilateral snake. Introduction The graphs which are used here are finite, undirected graphs. Here V(G) indicates vertices and E(G) indicates edges. For all described view of Graph Labeling we refer to J.A. Gallian [] and we follow Harary [] for all other standard terminology and notations in Graph

2 S. S. Sandhya et al. Theory. We will provide short summary and definitions which are useful for the present investigation. The notion of Geometric mean Labeling has introduced by S. Somasundaram, Rand Ponraj, P. Vidhyarani []. S. Somasundaram and R. Ponraj introduced Mean Labeling []. The notion of Harmonic Mean Labeling has introduced by S. Somasundaram, R. Ponraj and S.S. Sandhya []. C. Jeyasekaran, S.S. Sandhya and C. David Raj has introduced Super Harmonic Mean Labeling []. S.S. Sandhya, E. Ebin Raja Merly and G.D. Jemi has introduced Super Heronian Mean Labeling. 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, f*(e)= f(u)+ f(u)f(v) +f(v) [OR] f(u)+ f(u)f(v) +f(v) 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. Theorem:. Paths are Super Heronian Mean Graph. Theorem:. Ladders are Super Heronian Mean Graph.. Main Results Theorem:. Triangular snakes are Super Heronian mean graphs. Proof: Let Tn be a Triangular snake which is obtained from a Path Pn=uu... un by joining ui to ui+ to a new vertex vi, i n-. Define a function f: V(Tn) {,,...,p+q} by, f(ui)=i- ; i n f(vi)=i- ; i n- f(uiui+)=i- ; i n-; f(uivi)=i- ; i n- f(viui+)=i; i n- Then we get distinct edge labels. Hence Triangular snakes are Super Heronian mean graphs.

3 Some results on super Heronian mean labeling of graphs 7 Example:. A Super Heronian mean labeling of T is given below Figure: Theorem:. Quadrilateral snakes are Super Heronian mean graph. Proof: Let Qn be a Quadrilateral snakes which is obtained from a Path Pn=uu... un by joining ui and ui+ to a new vertex vi and wi respectively and joining vi and wi. Define a function f: V(Qn) {,,...,p+q} by, f(ui)=7i- ; i n f(vi)=7i- ; i n- f(wi)=7i- ; i n- f(uiui+)=7i- ; i n- f(uivi)=7i- ; i n- f(viwi)=7i- ; i n- f(wiui+)=7i ; i n- Then the labels of the edges are distinct. Hence Qn are Super Heronian mean graphs. Example:. A Super Heronian mean labeling of Q is given below. v w u 9 Figure:

4 S. S. Sandhya et al. Theorem:. Alternate Triangular snakes A(Tn) are Super Heronian mean Graphs. Proof: Let G be a graph A(Tn). Consider a path uu... un. To construct G, join ui and ui+ alternatively with a new vertex vi. Here we consider two different cases. Case (i): If the alternate Triangular snake A(Tn) starts from u, then we need to consider two subcases. Subcase (i)(a): If n is odd, then Define a function f: V(G) {,,...,p+q} by, f(ui-)=7i- ; i ( n )+ f(ui)=7i- ; i ( n ) f(v)= f(vi)=7i- ; i ( n )+ f(uu)= f(ui-ui)=7i- ; i ( n ) f(uiui+)=7i ; i ( n ) f(viui-)=7i- ; i ( n ) f(viui)=7i- ; i ( n ) The labeling pattern of A(T) is shown below. 0 7 V V 9 9 U U 9 9 Figure:

5 Some results on super Heronian mean labeling of graphs 9 Then the edge labels are distinct. Hence f provides a Super Heronian mean labeling of G. Subcase (i) (b): If n is even, then Define a function f: V(G) {,,...,p+q} by, f(ui-)=7i- ; i n f(ui)=7i- ; i n f(v)=, f(vi)=7i- ; i n f(uu)=, f(ui-ui)=7i- ; i n f(uiui+)=7i ; i ( n ) f(viui-)=7i- ; i n f(viui)=7i- ; i n Then we get distinct edge labels. The labeling pattern of A(T)is displayed below V 0 7 V 9 9 U U Figure: Hence f provides a Super Heronian mean labeling of G. Case (ii): If A(Tn) starts from u, then we need two subcases. Subcase (ii)(a): If n is odd, then

6 90 S. S. Sandhya et al. Define a function f: V(G) {,,...,p+q} by, f(ui-)=7i- ; i ( n )+ f(ui)=7i- ; i ( n ) f(vi)=7i- ; i ( n ) f(ui-ui)=7i- ; i ( n ) f(uiui-)=7i- ; i ( n ) f(viui)=7i- ; i ( n ) f(viui+)=7i ; i ( n ) Then we get distinct edge labels. The labeling pattern of A(T) is shown below. V 0 V 7 U U 7 Figure: Hence f provides a Super Heronian mean labeling of G. Subcase (ii)(b): If n is even, then Define a function f: V(G) {,,...,p+q} by, f(ui-)=7i- ; i n f(ui)=7i- ; i n f(vi)=7i- ; i ( n )

7 Some results on super Heronian mean labeling of graphs 9 f(ui-ui)=7i- ; i n f(uiui+)=7i- ; i ( n ) f(viui)=7i- ; i ( n ) f(viui+)=7i ; i ( n ) The labeling pattern of A(T) is shown below. V 0 V 7 U U Figure: Theorem:. Alternate Quadrilateral snakes A(Qn) are Super Heronian mean Graphs. Proof: Let G be the graph A(Qn).Consider a Path uu... un. To construct G, Join ui and ui+ alternatively with two new vertices vi and wi respectively. There are two different cases to be considered. Case (i): If alternate Quadrilateral snake A(Qn) starts from u,then we need to consider two sub cases. Sub case (i)(a): If n is odd, then Define a function f: V(G) {,, ,p+q} by, f(u)=, f(ui-)=9i- ; i ( n )+ f(ui)=9i- ; i ( n ) f(v)=, f(vi)=9i- ; i ( n ) f(w)=, f(wi)=9i- ; i ( n )

8 9 S. S. Sandhya et al. f(uu)=7, f(ui-ui)=9i- ; i ( n ) f(uiui+)=9i ; i ( n ) f(uv)=, f(ui-vi)=9i-7 ; i ( n ) f(vw)=, f(viwi)=9i- ; i ( n ) f(wu)=, f(wiui)=9i- ; i ( n ) The labeling pattern of A(Q) is displayed below Figure: 7 This makes f is a Super Heronian mean labeling of G. Sub case(i) (b): If n is even, then Define a function f: V(G) {,, ,p+q} by, f(u)=, f(ui-)=9i- ; i n f(ui)=9i- ; i n f(v)=, f(vi)=9i- ; i n f(w)=, f(wi)=9i- ; i n f(uu)=7, f(ui-ui)=9i- ; i n f(uiui+)=9i ; i ( n )

9 Some results on super Heronian mean labeling of graphs 9 f(uv)=, f(uivi)=9i-7 ; i n f(vw)=, f(viwi)=9i- ; i n f((wu)=, f(wiui)=9i- ; i n The labeling pattern of A(Q) is shown below Figure: Case (ii): If the Alternate Quadrilateral snake A(Qn) starts from u, then we consider two sub cases. Sub case(ii)(a): If n is odd, then Define a function f: V(G) {,, ,p+q} by, f(ui-)=9i- ; i ( n ) f(ui)=9i- ; i ( n ) f(v)=, f(vi)=9i- ; i ( n ) f(wi)=9i- ; i ( n ) f(ui-ui)=9i-7 ; i ( n ) f(uu)=, f(uiui+)=9i- ; i ( n ) f(uivi)=9i- ; i ( n )

10 9 S. S. Sandhya et al. f(ui+wi)=9i ; i ( n ) f(viwi)=9i- ; i ( n ) The labeling pattern of A(Q) is displayed below. From the above labeling pattern, f provides a Super Heronian mean labeling of G Figure: 9 Sub case: (ii) (b): If n is even, then Define a function f: V(G) {,, ,p+q} by, f(ui-)=9i- ; i n f(ui)=9i- ; i n f(v)=, f(vi)=9i- ; i ( n ) f(wi)= 9i-l ; i ( n ) f(ui-ui)=9i-7 ; i n f(uu)=, f(uiui+)=9i- ; i ( n ) f(uivi)=9i- ; i ( n ) f(ui+wi)=9i ; i ( n ) f(viwi)=9i- ; i ( n ) Then we get distinct edge labels. The labeling pattern of A(Q)is shown below.

11 Some results on super Heronian mean labeling of graphs Figure: 0 This makes f is a Super Heronian mean labeling of G.From all the above cases, we conclude that Alternate Quadrilateral snakes A(Qn) are Super Heronian mean graphs. References [] J.A. Gallian, A Dynamic Survey of Graph labelling, The Electronic Journal of Combinatorics, (0). [] F. Harary, Graph Theory, Narosa Publishing House Reading, New Delhi, 9. [] S. Somasundaram and R. Ponraj, Mean Labeling of Graphs, National Academy of Science Letters, 0-. [] S. Somasundaram, R. Ponraj and S. S. Sandhya, Harmonic Mean Labeling of Graphs communicated, Journal of Combinatorial Mathematics and Combinatorial Computing. [] C. Jeyasekaran and C. David Raj, Some Results on Super Harmonic Mean Graphs, International Journal of Mathematics Trend and Tecnology, (0), no., -. [] S. Somasundaram, R. Ponraj and P. Vidhyarani, Geometric Mean Labeling of Graphs, Bulletin of Pure and Applied Sciences, 0E (0), -0. [7] S. S. Sandhya, E. Ebin Raja Merly and G. D. Jemi, Super Heronian Mean Labeling of Graphs, communicated to International Journal of Mathematical Forum. Received: August, 0; Published: December, 0