Heronian Mean Labeling of. Disconnected Graphs

Transcription

1 International Journal of Contemporary Mathematical Sciences Vol. 12, 2017, no. 5, HIKARI Ltd, Heronian Mean Labeling of Disconnected Graphs S.S. Sandhya Department of Mathematics Sree Ayyappa College for Women Chunkankadai , Tamilnadu, India E. Ebin Raja Merly Department of Mathematics Nesamony Memorial Christian College Marthandam , Tamilnadu, India S.D. Deepa Nesamony Memorial Christian College Marthandam , Tamilnadu, India Copyright 2017 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 In this paper, we contribute some new results for Heronian Mean labeling of graphs. We prove that disconnected Heronian Mean Graphs are Heronian Mean Graphs. We use some standard graphs to derive the results for disconnected graphs. Mathematics Subject Classification: 05C78 Keywords: Graph, Heronian Mean Graph, Path, Cycle, Comb, Ladder, Triangular Snake, Quadrilateral Snake

2 202 S.S. Sandhya et al. 1. Introduction By a graph we mean a finite undirected graph without loops or parallel edges. For all detailed survey of graph labeling, we refer to J.A. Gallian [1]. For all other standard terminology and notations we follow Harary [2]. The concept of Mean labeling has been introduced by S. Somasundaram and R. Ponraj [3] in S. Somasundaram and S.S. Sandhya introduced Harmonic mean labeling [4] in Motivated by the above works we introduced a new type of labeling called Heronian Mean Labeling in [5]. In this paper we investigate the Heronian Mean Labeling of some disconnected graphs. We will provide brief summary of definitions and other information which are necessary for our present investigation. A Path P n is a walk in which all the vertices are distinct. A Cycle C n is a Closed Path. The graph obtained by joining a single pendant edge to each vertex of a Path is called a Comb. The corona G 1 ʘ G 2 is defined as the graph G obtained by taking one copy of G 1 (which has P 1 vertices) and P 1 copies of G 2 and then joining the i th vertex of G 1 to every vertices in the i th copy of G 2. The graph C n ʘ K 1 is called crown. The Ladder L n is the product graph P 2 P n. A Triangular Snake T n is obtained from a path u1,u2,.un by joining ui and ui+1 to a new vertex vi for 1 i n 1. That is every edge of a path is replaced by a triangle C 3. A Quadrilateral Snake Q n is obtained from a path u1,u2,.un by joining ui and ui+1 to two new vertices vi and wi respectively and then joining vi and wi. That is every edge of a path is replaced by a cycle C 4. Definition 1.1: A graph G=(V,E) with p vertices and q edges is said to be a Heronian Mean graph if it is possible to label the vertices x V with distinct labels f(x) from 1,2,,q+1 in such a way that when each edge e = uv is labeled with, f(u) + f(u)f(v) + f(v) f(u) + f(u)f(v) + f(v) f(e = uv) = (OR) 3 3 Then the edge labels are distinct. In this case f is called a Heronian Mean labeling of G. Theorem 1.2: Any Path P n is a Heronian mean graph. Theorem 1.3: Any Comb P n ʘK 1 is a Heronian mean graph. Theorem 1.4: Any Cycle C n is a Heronian mean graph. Theorem 1.5: Crown, C n ʘK 1 is a Heronian mean graph for all n 3. Theorem 1.6: Any Triangular Snake T n is a Heronian mean graph. Theorem 1.7: Any Quadrilateral Snake Q n is a Heronian mean graph. Theorem 1.8: Any Ladder L n is a Heronian mean graph.

3 Heronian mean labeling of disconnected graphs Main Results Theorem: 2.1 P m P n is a Heronian mean graph. Let Pm be a path u1u2u3.um and Pn be a path v1v2v3.vn Define a function f: V(P m P n ) {1,2,3,.., q + 1} by f(u i ) = i, 1 i m. Edges are labeled with, f(u i u i+1 ) = i, 1 i m, f(v i v i+1 ) = i, Hence P m P n is a Heronian mean graph. f(v i ) = m + i, Example 2.2: A Heronian mean labeling of P 5 P 4 is given below. Figure: 1 Theorem: 2.3 C m P n is a Heronian mean graph for m 3 and n > 1. Let C m be the cycle u1u2u3.umu1 and Pn be a path v1v2v3.vn Define a function f: V(C m P n ) {1,2,3,, q + 1} by f(u i ) = i, 1 i m. Edges of C m are labeled by f(u i u i+1 ) = i + 1, 1 i m, f(u m u 1 ) = 1 Edges of P n are labeled by {m + 1, m + 2, m + n 1}. Hence C m P n is a Heronian mean graph if m 3 and n > 1. f(v i ) = m + i, Example 2.4: A Heronian mean labeling of C 5 P 4 is given below.

4 204 S.S. Sandhya et al. Figure: 2 Theorem: 2.5 C m (P n ʘ K 1 ) is a Heronian mean graph for m 3 and n > 1. Let C m be the cycle u1u2u3.umu1 and P n ʘ K 1 be a graph obtained from a path v 1 v 2. v n by joining the vertex v i to pendant vertices w i Define a function, f: V(C m (P n ʘ K 1 ) ) {1,2,3,. q + 1} by f(u i ) = i, 1 i m f(v i ) = m + (2i 1), 1 i n f(w i ) = m + 2i, We get distinct edge labels for C m. Edges of P n ʘ K 1 are labeled by {m + 1, m + 2, m + 2n 1}. Hence C m (P n ʘ K 1 ) is a Heronian mean graph if m 3 and n > 1. Example 2.6: A Heronian mean labeling of C 4 (P 5 ʘ K 1 ) is given below. Figure: 3 Theorem: 2.7 (C m ʘ K 1 ) P n is a Heronian mean graph for m 3 and n > 1. Let C m ʘ K 1 be the cycle u1u2u3.umu1 by joining the vertex u i to pendant vertices v i and let Pn be a path w1w2w3.wn Define a function, f: V((C m ʘ K 1 ) P n ) {1,2,3,. q + 1} by f(u i ) = 2i, 1 i m

5 Heronian mean labeling of disconnected graphs 205 f(v i ) = 2i 1, f(w i ) = m + i, 1 i m We get distinct edge labels for C m ʘ K 1. Edges of P n are labeled by {2m + 1,2m + 2, 2m + n 1}. Hence (C m ʘ K 1 ) P n is a Heronian mean graph if m 3 and n > 1. Example 2.8: A Heronian mean labeling of (C 3 ʘ K 1 ) P 6 is given below. Figure: 4 Theorem: 2.9 C m (C n ʘ K 1 ) is a Heronian mean graph for m 3 and n 3. Let C m be the cycle u1u2u3.umu1 and C n ʘ K 1 be a graph obtained from a path v 1 v 2. v n by joining the vertex v i to pendant vertices w i Define a function, f: V(C m (C n ʘ K 1 ) ) {1,2,3,. q + 1} by f(u i ) = i, 1 i m f(v i ) = m + 2i, 1 i n f(w i ) = m + (2i 1), We get distinct edge labels for C m and C n ʘ K 1. Hence C m (C n ʘ K 1 ) is a Heronian mean graph if m 3 and n 3. Example 2.10: A Heronian mean labeling of C 4 (C 3 ʘ K 1 ) is given below. Figure: 5

6 206 S.S. Sandhya et al. Theorem: 2.11 C m L n is a Heronian mean graph for m 3 and n > 1. Let C m be the cycle u1u2u3.umu1 and L n be a ladder connecting two paths v 1 v 2. v n and w 1 w 2. w n. Define a function, f: V(C m L n ) {1,2,3,. 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), We get distinct edge labels for C m. Edges of L n are labeled by {m + 1, m + 2, m + 3n 2}. Hence C m L n is a Heronian mean graph if m 3 and n > 1. Example 2.12: A Heronian mean labeling of C 4 L 5 is given below. Figure: 6 Theorem: 2.13 (C m ʘ K 1 ) L n is a Heronian mean graph for m 3 and n > 1. Let C m ʘ K 1 be a graph obtained from the cycle u1u2u3.umu1 by joining the vertex u i to pendant vertices v i and let Ln be a ladder connecting two paths x 1 x 2. x n and y 1 y 2. y n Define a function, f: V((C m ʘ K 1 ) L n ) {1,2,3,. q + 1} by f(u i ) = 2i, 1 i m f(v i ) = 2i 1, 1 i m f(x i ) = m + (3i 2), f(y i ) = m + (3i 1), 1 i n We get distinct edge labels for C m ʘ K 1. Edges of L n are labeled by {2m + 1,2m + 2, 2m + 3n 2}. Hence (C m ʘ K 1 ) L n is a Heronian mean graph if m 3 and n > 1.

7 Heronian mean labeling of disconnected graphs 207 Example 2.14: A Heronian mean labeling of (C 3 ʘ K 1 ) L 5 is given below. Figure: 7 Theorem: 2.15 (C m ʘ K 1 ) (P n ʘ K 1 ) is a Heronian mean graph for m 3 and n > 1. Let C m ʘ K 1 be a graph obtained from the cycle u1u2u3.umu1 by joining the vertex u i to pendant vertices v i and let P n ʘ K 1 be a graph obtained from a path x 1 x 2. x n by joining the vertex x i to pendant vertices y i Define a function, f: V((C m ʘ K 1 ) (P n ʘ K 1 ) ) {1,2,3,. q + 1} by f(u i ) = 2i, 1 i m f(v i ) = 2i 1, 1 i m f(x i ) = m + (2i 1), f(y i ) = m + 2i, 1 i n We get distinct edge labels for C m ʘ K 1. Edges of P n ʘ K 1 are labeled by {2m + 1,2m + 2, 2m + 2n 1}. Hence (C m ʘ K 1 ) (P n ʘ K 1 ) s a Heronian mean graph if m 3 and n > 1. Example 2.16: A Heronian mean labeling of (C 3 ʘ K 1 ) (P 5 ʘ K 1 ) below. is given Figure: 8

8 208 S.S. Sandhya et al. 3. Conclusion The Study of labeled graph is important due to its diversified applications. It is very interesting to investigate disconnected graphs which admit Heronian Mean Labeling. The derived results are demonstrated by means of sufficient illustrations which provide better understanding. It is possible to investigate similar results for several other graphs. Acknowledgements. The authors are thankful to the referee for their valuable comments and suggestions. References [1] J.A. Gallian, A Dynamic Survey of Graph Labeling, The Electronic Journal of Combinatorics, (2013). [2] F. Harary, Graph Theory, Narosa Publishing House, New Delhi, [3] S. Somasundaram and R. Ponraj, Mean Labeling of graphs, National Academy of Science Letters, 26 (2003), [4] S. Somasundaram, R. Ponraj and S.S. Sandhya, Harmonic Mean Labeling of Graphs, Journal of Combinatorial Mathematics and Combinatorial Computing, to appear. [5] S.S. Sandhya, E. Ebin Raja Merly and S.D. Deepa, Heronian Mean Labeling of Graphs, International Mathematical Forum, 12 (2017), no. 15, [6] S.S. Sandhya, E. Ebin Raja Merly and S.D. Deepa, Some Results on Heronian Mean Labeling of Graphs, Journal of Discrete Mathematical Sciences and Cryptography, to appear. Received: August 17, 2016; Published: July 17, 2017