Root Square Mean Labeling of Some More. Disconnected Graphs

Transcription

1 International Mathematical Forum, Vol. 10, 2015, no. 1, HIKARI Ltd, Root Square Mean Labeling of Some More Disconnected Graphs S. S. Sandhya Department of Mathematics SreeAyyappa College for Women Chunkankadai , India S.Somasundaram Department of Mathematics ManonmaniamSundaranar University Tirunelveli , India S.Anusa Department of Mathematics Arunachala College of Engineering for Women Vellichanthai , India Copyright 2014 S. S. Sandhya, S. Somasundaram and S. Anusa. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use,distribution, and reproduction in any medium, provided the original work is properly cited. Abstract A graph G = (V, E) with p vertices and q edges is called a Root Square Mean graph if it is possible to label the vertices xεv with distinct labels f(x) from 1,2,, q + 1 in such a way that when each edge e = uv is labeled with f(e = uv) = f(u)2 +f(v) 2 or f(u)2 +f(v) 2, then the edge labels are distinct. 2 2 In this case f is called a Root Square Mean labeling of G. In this paper we prove that some disconnected graphs are Root Square Mean graphs. Keywords: Graph, Root Square Mean labeling, Path, Cycle, Comb, Ladder

2 26 S. S. Sandhya, S. Somasundaram and S. Anusa 1. Introduction The graph considered here are simple, finite and undirected graph G = (V, E) with p vertices and q edges. For a detailed survey of graph labeling we refer to Gallain [1]. For all other standard terminology and notations we follow Harary [2]. S.S.Sandhya,S.Somasundaram and S.Anusa introduced the concept of Root Square Mean labeling of graphs in [4] and studied their behavior in [5],[6], and [7].In this paper we investigate the Root Square mean labeling of some disconnected graphs. We now give the definitions which are useful for the present study. Definition1.1: A graph G = (V, E) with p vertices and q edges is called a Root Square Mean graph if it is possible to label the vertices xεv with distinct labels f(x) from 1,2,, q + 1 in such a way that when each edge e = uv is labeled with f(e = uv) = f(u)2 +f(v) 2 or f(u)2 +f(v) 2, then the edge labels are distinct. In 2 this case f is called a Root Square Mean labeling of G. Definition1.2: 2 The union of two graphs G 1 = (V 1, E 1 ) and G 2 = (V 2, E 2 ) is a graph G = G 1 G 2 with vertex set V = V 1 V 2 and the edge set = E 1 E 2. Definition1.3: The Corona of two graphs G 1 and G 2 is the graph G = G 1 G 2 formed by taking one copy of G 1 and V(G 1 ) copies of G 2 where the i th vertex of G 1 is adjacent to every vertex in the i th copy of G 2. 2.Main Results Theorem2.1:C m (P n K 1 ) is a Root Square Mean graph. Let u 1 u 2 u m u 1 be the cycle C m. Let v 1 v 2 v n be the path P n and let w i be the vertex which is joined to the vertex v i, 1 i n of the path P n. Let G = C m (P n K 1 ). Define a function f: V(G) {1,2,, q + 1} by f(u i ) = i, 1 i m f(v i ) = m + 2i 1, 1 i n f(w i ) = m + 2i, 1 i n

3 Root Square mean labeling of some more disconnected graphs 27 Example2.2: The Root Square Mean labeling of C 8 (P 5 K 1 ) is given below. Theorem2.3: Figure1 (C m K 1 ) (P n K 1 ) is a Root Square Mean graph. Let C m be the cycle u 1 u 2 u m u 1 and let v i be the pendent vertex attached to u i, 1 i m.letp n = w 1 w 2 w n be the path on n vertices. Join a vertex t i to w i, 1 i n. LetG = (C m K 1 ) (P n K 1 ). Define a function f: V(G) {1,2,, q + 1} by f(u i ) = 2i 1, 1 i m f(v i ) = 2i, 1 i m f(w i ) = 2m + 2i 1, 1 i n f(t i ) = 2m + 2i, 1 i n Example2.4: The Root Square Mean labeling of(c 6 K 1 ) (P 5 K 1 ) is given below.

4 28 S. S. Sandhya, S. Somasundaram and S. Anusa Figure2 Theorem2.5:(C m K 2 ) (P n K 1 ) is a Root Square Mean graph. Let u 1 u 2 u m u 1 be the cycle C m and let v i, w i be the vertices which are joined to the vertex u i, 1 i m of the cycle C m. Let s 1 s 2 s n be the path P n and let t i be the vertex which is joined to the vertex s i, 1 i n, of the path P n. Let G = (C m K 2 ) (P n K 1 ). Define a function f: V(G) {1,2,, q + 1} by f(u i ) = 3i 1, 1 i m f(v i ) = 3i 2, 1 i m f(w i ) = 3i, 1 i m f(s i ) = 3m + 2i 1, 1 i n f(t i ) = 3m + 2i, 1 i n Example2.6: The Root Square Mean labeling of(c 6 K 2 ) (P 6 K 1 ) is given below.

5 Root Square mean labeling of some more disconnected graphs 29 Theorem2.7: C m L n is a Root Square Mean graph. Figure3 Let u 1 u 2 u m u 1 be the cycle C m.let L n be the Ladder graph with vertices v i and w i, 1 i n. Let G = C m L n. Define a function f: V(G) {1,2,, q + 1} by f(u i ) = i, 1 i m f(v i ) = m + 3i 2, 1 i n f(w i ) = m + 3i 1, 1 i n Example2.8: The Root Square Mean labeling ofc 7 L 6 is given below. Figure4

6 30 S. S. Sandhya, S. Somasundaram and S. Anusa Theorem2.9: (C m K 1 ) L n is a Root Square Mean graph. Let u 1 u 2 u m u 1 be the cycle C m and let v i be the vertex which is joined to the vertex u i, 1 i m of the cycle C m. Let x i and y i, 1 i nbe the vertices of L n. Let G = (C m K 1 ) L n.define a function f: V(G) {1,2,, q + 1} by f(u i ) = 2i 1, 1 i m f(v i ) = 2i, 1 i m f(x i ) = 2m + 3i 1, 1 i n f(y i ) = 2m + 3i 2, 1 i n Example2.10: The Root Square Mean labeling of(c 9 K 1 ) L 6 is given below. Figure5 Theorem2.11:C n (P m K 3 ) is a Root Square Mean graph. Let C n be the cycle u 1 u 2 u n u 1. Let v 1 v 2 v m be the path P m. Let x i, y i, 1 i m be the vertices of K 3 which are attached to the vertices of P m. Let G = C n (P m K 3 ). Define a function f: V(G) {1,2,, q + 1} by f(u i ) = i, 1 i n f(v i ) = n + 4i 3, 1 i m f(x i ) = n + 4i 2, 1 i m f(y i ) = n + 4i 1, 1 i m

7 Root Square mean labeling of some more disconnected graphs 31 Example2.12: The Root Square Mean labeling ofc 9 (P 5 K 3 )is given below. Theorem2.13: Figure6 (C n K 1 ) (P m K 3 ) is a Root Square Mean graph. Let C n be the cycle u 1 u 2 u n u 1 and let v i be the vertex which is joined to the vertex u i, 1 i n of the cycle C n. Let w 1 w 2 w m be the path P m. Let x i, y i be the vertices of K 3 which are attached to each vertex of P m. Let G = (C n K 1 ) (P m K 3 ). Define a function f: V(G) {1,2,, q + 1} by f(u i ) = 2i 1, 1 i n f(v i ) = 2i, 1 i n f(w i ) = 2n + 4i 3, 1 i m f(x i ) = 2n + 4i 2, 1 i m f(y i ) = 2n + 4i 1, 1 i m Then the edge labels are distinct. Hence f is a Root Square Mean Labeling ofg. Example2.14: The Root Square Mean labeling of(c 8 K 1 ) (P 4 K 3 )is given below.

8 32 S. S. Sandhya, S. Somasundaram and S. Anusa Figure7 Theorem2.15: (C n K 3 ) P m is a Root Square Mean graph. Let u 1 u 2 u n u 1 be the cycle C n. Let v i, w i, 1 i nbe the vertices of K 3 which are attached to the vertices of C n. Let t i, 1 i m be the vertices of the path P m. Let G = (C n K 3 ) P m. Define a function f: V(G) {1,2,, q + 1} by f(u 1 ) = 3, f(u i ) = 4i 2, 2 i n f(v i ) = 4i 3, 1 i n f(w 1 ) = 2, f(w i ) = 4i,2 i n f(t i ) = 4n + i, 1 i m Example2.16: The labeling pattern of(c 6 K 3 ) P 5 is given below. Figure8

9 Root Square mean labeling of some more disconnected graphs 33 Theorem2.17: (C n K 3 ) (P m K 3 ) is a Root Square Mean graph. Let u 1 u 2 u n u 1 be the cycle C n. Let v i, w i, 1 i n be the vertices of K 3 which are attached to the vertices of C n. Let t 1 t 2 t m be the path P m.let x i, y i be the vertices of K 3 which are attached to t i, 1 i m. Let G = (C n K 3 ) (P m K 3 ). Define a function f: V(G) {1,2,, q + 1} by f(u 1 ) = 3, f(u i ) = 4i 2, 2 i n f(v i ) = 4i 3, 1 i n f(w 1 ) = 2, f(w i ) = 4i,2 i n f(t i ) = 4n + 4i 3, 1 i m f(x i ) = 4n + 4i 2, 1 i m f(y i ) = 4n + 4i 1, 1 i m Example2.18: The labeling pattern of(c 6 K 3 ) (P 5 K 3 )is given below. Figure9 References [1] Gallian. J.A, 2012, A dynamic survey of graph labeling. The electronic Journal of Combinatories17#DS6. [2] HararyF, 1988, Graph Theory, Narosa Publishing House Reading, New Delhi. [3] Sandhya S. S, Somasundaram S, Geometric Mean Labeling of Disconnected Graphs Future Prospects in Multi Disciplinary Research. ISBN page no:134 to 136.

10 34 S. S. Sandhya, S. Somasundaram and S. Anusa [4] Sandhya S. S, Somasundaram S, AnusaS, Root Square Mean Labeling of Graphs, submitted to International Journal of Contemporary Mathematical Sciences. [5] Sandhya S. S, Somasundaram. S, AnusaS, Some Results on Root Square Mean Graphs, submitted to Journal of Scientific Research. [6] Sandhya S. S, Somasundaram S, AnusaS, Some More Results on Root Square Mean Graphs, submitted to Journal of Mathematics Research. [7] Sandhya S. S, Somasundaram.S, AnusaS, Further Results on Root Square Labeling, submitted to Bulletin of Pure and Applied Mathematics Sciences. Received: December 5, 2014; Published: January 5, 2015