Root Cube Mean Labeling of Graphs

Transcription

1 International Journal of Engineering Science, Advanced Computing and Bio-Technology Vol 8, No 4, October December 07, pp Root Cube Mean Labeling of Graphs R Gowri and *G Vembarasi Department of mathematics Government College for women(a) kumbakonam, India gowrigck@rediffimailcom, *vembagopal@gmailcom Abstract: A graph = (,) with vertices and edges is said to be a Root Cube Mean graph if it is possible to label the vertices with distinct elements () from,,,+ in such a way that when each edge = is labeled with f(e=uv) = or, then the resulting edge labels are distinct Here is called a Root Cube Mean labeling of In this paper we prove that Path, Cycle, Comb, Ladder, and Triangular Snake are Root Cube Mean graphs Keywords: Mean Labeling of graphs, Root Cube Mean Labeling of graphs AMS Subject Classification (00):05C78 Introduction Graceful and Harmonious Labelings, Variations of Graceful Labelings are studied by Gallian[] The concept of Mean Labeling has been introduced by Somasundaram and Ponraj [] Root Square Mean Labeling of Graphs has been introduced by Sandhya, Somasundaram and Anusa[4] Motivated by the above works we introduced a new type of labeling called Root Cube Mean Labeling The standard terminology and notations we follow Harary [] In this paper we investigate the Root Cube Mean Labeling of Path, Cycle, Comb, Ladder, and Triangular Snake Graphs Preliminaries Definition : [4] A walk in which vertices are distinct is called a path A path on n vertices is denoted by Definition : [4] A closed path is called a cycle A cycle on n vertices is denoted by C n Definition : [4] The graph obtained by joining a single pendent edge to each vertex of a path is called a Comb graph pn Received: July, 07; Revised: November, 07; Accepted: November, 07 *Corresponding Author DOI: 06674/ijesacbt/07/4947

2 49 International Journal of Engineering Science, Advanced Computing and Bio-Technology Definition 4: [4] The product graph P X P n is called a Ladder and it is denoted by L n Definition 5: [4] A Triangular Snake T n is obtained from a path u, u,, un by joining u i and u i to a new vertex vi for i n That is every edge of a path is replaced by a triangle C Definition 6: Let G be simple Graphs with vertex set V (G) and the Edge set E (G) respectively Vertex set V (G) are labeled arbitrary by positive integers and let E (e) denoted the edge label such that it is the sum of labels of vertices incident with edge e Root Cube Mean Labeling of Graphs Definition : A graph G ( V, E) with p vertices and q edges is said to be a Root Cube Mean Graph if it is possible to label the vertices x v with distinct elements f (x) from,,, q in f (u) f (v) such a way that when each edge e uv is labeled with f (e uv) f (u) f (v) or, than the resulting edge labels are distinct Here f is called a Root Cube Mean Labeling of G Theorem : Any path p n is a Root Cube Mean Labeling Graph Let G pn be the graph with vertices u, u,, un and the edgese,e,eq Define the function f : V(pn) (,,,q ) as follows f(ui ) i, for i n and the induced edge labeling function f : E( G) N is defined by Now, f (e uv) f (ui ui ) f (u) f (u ) i f (v) f (u i)

3 Root Cube Mean Labeling of Graphs 50 i i i = for i n n n n Clearly f (un un ) Hence the edge labels are distinct Thus path graph admits a Root Cube Mean Labeling of Graphs Example : The following example shows that p 5 is Root Cube Mean Labeling Graph Figure Here, f (u ), f (u ), f (u ), f (u ) 4, f (u ) Now, 9 f (uu ), similarly f (uu) 4, f (u u ) 9 Hence Edge Labeling of p5 are distinct 4 5 f (u u ) 6, 4 Theorem 4: Any Cyclic C n is a Root Cube Mean Labeling Graph Let G Cn be the graph with verticesu,u,,un and the edgese,e,eq Define the function f : V(Cn) (,,,q ) as follows f(ui ) i for i n and the induced edge labeling function f : E(G) N defined by f (u) f (v) f (e uv) i i i Now, f (uiui ) for i n Clearly, f (u n n u ) n n n

4 5 International Journal of Engineering Science, Advanced Computing and Bio-Technology n f (un u) Thus the edge labels are distinct Hence the cycle graphc admits a Root Cube Mean Labeling of Graphs n Example 5: The following is an example forc 5 is a Root Cube Mean Labeling Graphs Figure Theorem 6: The Ladder L n is a Root Cube Mean Labeling Graph Let G be the Ladder graph with verticesu,u,,un and the edgese,e,eq Define the function f : V(Ln) (,,,q ) as follows f (ui ) i, f (vi ) i for i n And the induced edge labeling function f : E( G) N defined by f (u) f (v) f (u) f (v) f (e uv) (or) Now, f ( uu i i) 8i 6i (or) 8i 6i for i n Clearly f ( un un) 8n 4n 0n4 (or) 8n 4n 0n 4 f ( vv i i) 8i i i4 (or) 8i i i 4 for i n f v v n n n 8n n n 4 ( n n) 8 4 f i n (or) 6i i 6i 6i i 6i (ui vi ) (or) for

5 Root Cube Mean Labeling of Graphs 5 6n n 6n 6n n 6n f (un vn ) (or) Here the edge labels are distinct Hence the Ladder graph admits a Root Cube Mean Labeling of Graphs Example 7: The Root Cube Mean labeling of L 6 is given below Figure Here f ( u ), f (u ), f (u) 5,, f (un ) and f (v), f (v) 4, f (v) 6,, f (vn) And the edge labels are f (uu ), f (uu) 8, f (uu4 ) 5,, f (u5u6 ) and f (v v ) 6, f (v v ),, f (v 5v 6 ) 6, f (uv), f (uv ) 7,, f (u6v 6 ) all are distinct Theorem: 8: Comb graph is Root Cube Mean Labeling Graphs Let G be a Comb graph with verticesu,u,,un and the edgese,e,eq Let pn be the path u,u,,un in G and join a vertex vi tou i,for i n Define the function f : V(Cn) (,,,q ) as follows f(ui ) i for i n and the induced edge labeling function f : E(G) N defined by f (u) f (v) f (e uv) Now, f ( uu i i) 8i 6i for i n 9

6 5 International Journal of Engineering Science, Advanced Computing and Bio-Technology Clearly f ( un un) 8n 4n 0n4 6i i 6i f (ui vi ) for i n 6n n 6n f (un vn ) Thus the edge labels are distinct Hence the Comb graph admits a Root Cube Mean Labeling of Graphs Example 9: Root Cube Mean Labeling of Comb graph obtained from P 6 is given below Figure 4 Here the vertex labels f (u), f (u ), f (u) 5,, f (un) and f (v), f (v) 4, f (v) 6,, f (vn), and the edge labels are f (u u ), f (u u ) 8, f (u u ) 5,, f (u u ), f (u v ), 4 f (uv ) 6,, f (u 6v 6 ) 9 Here the edge labels are distinct Hence Comb graph obtained from P 6 is Root Cube Mean Labeling of Graphs Theorem 0: Triangular Snake T n is a Root Cube Mean Labeling Graph Let T n be a triangular snake Define a function f : V(Tn ) (,,,q ) as follows f (ui ) i, i n, f (vi ) i, i n and the induced edge labeling function f : E(G) N is defined by 5 6

7 Root Cube Mean Labeling of Graphs 54 f (u) f (v) f (e uv) f (ui ) f (vi ) Now, f (ui ui ) f ( uu i i) 8i 6i for i n Clearly f ( un un) 8n 4n 0n4 6i i 6i f (ui vi ) for i n 6n n 6n f (un vn ) 6i i 6i f (ui vi ) for i n 6n 6n f (un vn ) Hence the edge labels are distinct Thus the Triangular SnakeT n is a Root Cube Mean Labeling of Graphs Example: : This example shows thatt 4 is Root cube Mean Labeling Graphs Figure 5 Conclusion: It is very interesting to investigate graphs which admit Root Cube Mean Labeling In this paper we proved that path, Cycle, Comb, Ladder, Triangular Snake is Root Cube Mean Labeling Graphs It is possible to investigate similar results for several other graphs

8 55 International Journal of Engineering Science, Advanced Computing and Bio-Technology References: [] Gallian, A dynamic Survey of graph labeling The electronic Journal of Combinatories7#DS6 (00) [] FHarary, Graph Theory, Narosa Publishing House Reading, New Delhi(988) [] RPonraj and SSomasundaram, Mean labeling of graphs, National Academy of Science Letters 6, (0), 0- [4] SSandhya, SSomasundaram and SAnusa,Root Square Mean Labeling of Graphs International Journal of Contemporary Mathematics Sciences9, (04), [5] SSandhya, SSomasundaram and SAnusa, Some More Results on Root Square Mean Graphs, Journal of Mathematics Research 7, No; (05), 7-8 [6] SSandhya, SSomasundaram and SAnusa, Root Square Mean Labeling of Some New Disconnected Graphs International Journal of Mathematics Trends and Technology5 (04), 85-9 [7] Sandhya, Somasundaram and Anusa, Root Square Mean Labeling of Subdivision of Some Graphs, Global Journal of Theoretical and Applied Mathematics Science 5, Number, (05), - Authors Profile: Dr R Gowri was born in Kumbakonam, Thanjavur, Tamil Nadu, India in 978 She received her BSc, MSc, and MPhil degree in Mathematics from Bharathidasan University, Tiruchirappalli, India She worked as a lecturer in Idhaya College for Women, Kumbakonam in the period of In 00, she joined as an Assistant Professor in the Srinivasa Ramanujan Research Centre, SASTRA University She did her research work in SASTRA University and received her PhD in 009 Then she joined as an Assistant Professor in Department of Mathematics, Government College for Women(A), Kumbakonam, Thanjavur, India in 0 Her areas of Interest are Topology, Graph Theory and Algebra She published more than 45 research papers in various International/National Journals Currently Seven Research Scholars are doing PhD under her guidance out of two them have submitted their PhD Thesis GG Vembarasi was born in Valangaiman, Thanjavur, TamilNadu, India in 984 She received her Bsc, MSc, and MPhil degree in Mathematics from the Bharathidasan University, Trichirappalli in 004, 007, and 008 respectivelyshe is doing her PhD under the guidance of Dr R Gowri, Assistant Professor, Department of Mathematics, Government College for Women(A), Kumbakonam Her main area of research is Root Cube Mean Labeling of Graphs