Geometric mean labeling of some more Disconnected Graphs
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1 Geometric mean labeling of some more Disconnected Graphs S 1.P.Viji, S 2. Somasundaram, S3.S.Sandhya, 1. Department of Mathematics, K.N.S.K. College of Engineering, Therekalputhoor Department of Mathematics, M.S. University, Tirunelveli Department of Mathematics, Sree Ayyappa College for Women, Chunkankadai: Abstract A Graph G = (V,E) with p vertices and q edges is said to be a Geometric mean graph if is possible to label the vertices x V with distinct labels f(x) from 1,2..q+1 is such a way that when each edge e=uv is labeled with f(e=uv) = f u f v (or) f u f(v), then the resulting edge labels are distinct. In this case f is called Geometric mean labeling of G. In this paper we prove that some disconnected graphs are Geometric Mean graphs. Key Words: Graph, Geometric Mean labeling, Path, Cycle, Comb, Ladder etc. 1.Introduction The graph considered here are finite and undirected graph G = (V, E)withp 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. Somasundaram and P. Vidhyarani introduced the concept of Geometric Mean labeling of graphs in [3] and studied their behavior in [4], [5], [6] and [7]. In this paper we investigate the Geometric mean labeling behavior of some disconnected graphs. The following definitions are useful for our present study. 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 Geometric Mean Proof: 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 vertexv i, 1 i nof the path P n. LetG = C m P n K 1. f u i = i, f v i = m + 2i 1, 1 i n f w i = m + 2i, 1 i n Example2.2: The Geometric Mean labeling of C 8 P 5 K 1 is given below. Definition1.1: A Graph G = (V,E) with p vertices and q edges is said to be a Geometric mean graph if is possible to label the vertices x V with distinct labels f(x) from 1,2..q+1 is such a way that when each edge e=uv is labeled with f(e=uv) = f u f(v) (or) f u f(v), then the resulting edge labels are distinct. In this case f is called Geometric mean labeling of G. Definition1.2: 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 = E 1 E 2 Figure1 ISSN: Page 1
2 Theorem2.3: C m K 1 P n K 1 is a Geometric Proof: Let C m be the cycleu 1 u 2 u m u 1 and let v i be the pendent vertex attached to u i,.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. f u i = 2i, 3 i m f v i = 2i 1, 3 i m f u i = 3i 1, f v i = 3i 2, f w i = 3i, f s i = 3m + 2i 1, 1 i n f t i = 3m + 2i, 1 i n From the above labeling pattern, we get distinct edge labels. Hencef is a Geometric Mean Labeling of G. Example2.6: The Geometric Mean labeling of C 6 K 2 P 6 K 1 is given below. f w i = 2m + 2i 1, 1 i n f t i = 2m + 2i, 1 i n Hence f is a Geometric Mean Labeling of G. Example2.4: The Geometric Mean labeling of C 6 K 1 P 5 K 1 is given below. Figure3 Theorem2.7:C m L n is a Geometric Figure2 Proof:Letu 1 u 2 u m u 1 be the cycle C m.let L n be the Ladder graph with vertices v i andw i, 1 i n.let G = C m L n. Theorem2.5: C m K 2 P n K 1 is a Geometric Proof: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, of the cycle C m. Let s 1 s 2 s n be the path P n and lett i be the vertex which is joined to the vertexs i (1 i n) of the path P n. LetG = C m K 2 P n K 1. f u i = i, f v i = m + 3i 2, 1 i n f w i = m + 3i 1, 1 i n Example2.8: The Geometric Mean labeling ofc 7 L 6 is given below. ISSN: Page 2
3 Theorem2.11:C m P n K 3 is a Geometric Mean Proof: Let C m be the cycle u 1 u 2 u m u 1. Let v 1 v 2 v n be the path P n. Let x i, y i, 1 i n be the vertices of K 3 which are attached to the vertices of P n. Let G = C m P n K 3. Figure4 Theorem2.9: C m K 1 L n is a Geometric Mean Proof:Let u 1 u 2 u m u 1 be the cycle C m and letv i be the vertex which is joined to the vertex u i, 1 i m of the cycle C m. Letx i andy i, 1 i nbe the vertices ofl n. Let G = C m K 1 L n. f u i = 2i, 3 i m f u i = i, f v i = m + 4i 2, 1 i n f x i = m + 4i 3, 1 i n f y i = m + 4i 1, 1 i n Example2.12: The Geometric Mean labeling ofc 9 P 5 K 3 is given below. f v i = 2i 1, 3 i m f x i = 2m + 3i 1, 1 i n f y i = 2m + 3i 2, 1 i n Example2.10: The Geometric Mean labeling of C 9 K 1 L 6 is given below. Figure6 Theorem2.13: C m K 1 P n K 3 is a Geometric Proof: Let C m be the cycleu 1 u 2 u m u 1 and let v i be the vertex which is joined to the vertex u i, 1 i m of the cycle C m. Let w 1 w 2 w n be the path P n. Let x i, y i be the vertices of K 3 which are attached to eachvertex of P m.let = C n K 1 P m K 3. Figure5 f u i = 2i, 3 i m ISSN: Page 3
4 f v i = 2i 1,3 i m f w i = 2m + 4i 2, 1 i n f x i = 2m + 4i 3, 1 i n f y i = 2m + 4i 1, 1 i n Hence f is a Geometric Mean Labeling ofg. Example2.14: The Geometric Mean labeling of C 8 K 1 P 4 K 3 is given below. Figure8 Theorem2.17: C m K 3 P n K 3 is a Geometric Proof:Let u 1 u 2 u m u 1 be the cycle C m. Let v i, w i, be the vertices of K 3 which are attached to the vertices ofc m. Let t 1 t 2 t n be the path P n.let x i, y i be the vertices of K 3 which are attachedto t i, 1 i n.let G = C m K 3 P n K 3. f u i = 4i 1, Figure7 Theorem2.15: C m K 3 P n is a Geometric Mean Proof: Let u 1 u 2 u m u 1 be the cycle C m. Let v i, w i, be the vertices of K 3 which are attached to the vertices of C n.let t i, 1 i n be the vertices of the path P n. Let G = C m K 3 P n. f u i = 4i 1, f v i = 4i 3, f v i = 4i 3, f w i = 4i, f t i = 4m + 4i 2, 1 i n f x i = 4m + 4i 3, 1 i n f y i = 4m + 4i 1, 1 i n Example2.18: The labeling pattern of C 6 K 3 P 5 K 3 is given below. f w i = 4i, f t i = 4m + i, 1 i n Example2.16: The labeling pattern of C 6 K 3 P 5 is given below. Figure9 ISSN: Page 4
5 REFERENCES: 1 Gallian. J.A, 2012, A dynamic Survey of graph labeling. The electronic Journal of Combinatories17#DS6. 2 Harary.F, 1988, Graph Theory, Narosa Publishing House Reading, New Delhi. [3]S. Somasundaram, P. Vidyarani and R. Ponraj Geometric Mean Labeling of Graphs, Bullettin of Pure and Applied Sciences. 30E(2)(2011), page [4]Sandhya.S.S, Somasundaram.S, Geometric Mean Labeling of Disconnected Graphs Future Prospects in Multi Disciplinary Research. ISBN page no:134 to 136. [5] S. Somasundaram. S.S. Sandhya, S.P Viji, Some New Results in Geometric Mean Graphs Kanyakumari Academy of Arts and Science. ISBN Vol. 3 page [6] S. Somasundaram. S.S. Sandhya, S.P Viji, A Note on Geometric Mean Graphs, International Journal of Mathematical Archive- 5(10)(2014), 1-8 ISSN [7] S. Somasundaram. S.S. Sandhya, S.P Viji, Few Results on Geometric Meam Graphs, International Journal of Mathematical Trends & Technology. ISSN Vol. 16 No. 1 (2014). [8] S. Somasundaram. S.S. Sandhya, S.P Viji, On Geometric Mean Graphs, communicated to International Journal of Mathematical Forum. ISSN: Page 5
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