GEOMETRIC MEAN LABELING OF SUBDIVISION ON SOME GRAPHS

Bulletin of Pure and Applied Sciences

Volume 34 E (Math & Stat.) Issue (No.1-2)2015: P.1-14

www.bpas.in DOI 10.5958/2320-3226.2015.00004.1 Print version ISSN 0970 6577 Online version ISSN 2320 3226

GEOMETRIC MEAN LABELING OF SUBDIVISION ON SOME GRAPHS

1 S. Somasundaram, ² S.S.Sandhya and ³ P. Viji

1 Department of Mathematics, M.S.University , Tirunelveli – 627012 Email: somutvl@gmail.com

2 Department of Mathematics, Sree Ayyappa College for Women Chunkankadai - 629003

Email: sssandhya2009@gmail.com

3 Department of Mathematics, K.N.S.K. College of Engineering Therekalputhoor – 629006.

Email: vijispillai84@gmail.com

Received on 05 August 2014: Accepted on 10 November 2015

ABSTRACT

A Graph G = (V, E) with p vertices and q edges is said to be a Geometric mean graph if it is possible to label the vertices x∈V with distinct labels f(x) from 1,2…..,q+1 in such way that when each edge e=uv is labeled with f(e=uv)= or , then the resulting edge labels are distinct. In this case, f is called Geometric mean labeling of G. In this paper, we investigate the Geometric mean labeling behaviour of subdivision on some standard graphs.

Keywords:Path, Cycle, Comb, Crown, Ladder, Geometric mean graph, Subdivision of Graphs.

1. INTRODUCTION

Throughout this paper we consider finite, undirected and simple graphs.

Let G be a graph with p vertices and q edges. For all terminologies and notations we follow [1]. There are several types of labeling and a detailed survey can be found in [2].

Subdivision of Mean labeling was introduced in [3]. The concept of Geometric mean

labeling was introduced in [5]. The Harmonic mean labeling was introduced in [4] and

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motivated the author to study the Geometric mean labeling of Subdivision on some standard graphs.

The following definitions are necessary for the present study.

Definition 1.1: A graph G= (V,E) with p vertices and q edges is said to be a Geometric mean 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(e=uv)= (or) ,

then the resulting edge labels are distinct. In this case f is called Geometric mean labeling of G.

Definition 1.2: A subdivision of a graph G is a graph resulting from the subdivision of edges in G. The subdivision of some edge e with end points {u,v}yields a graph containing one new vertex w and with an edge set replacing e by two new edges {u,w}

and {w,v}.

Definition1.3: A Path P _n is a walk in which all the vertices are distinct.

Definition 1.4: A Cycle C n is a closed walk in which no vertices repeated more than once.

Definition 1.5: Comb is a graph obtained by joining a single pendant edge to each vertex of a path P _n .

Definition 1.6: Any Cycle with a pendant edge attached at each vertex is called a Crown.

Definition 1.7: The graph P n AK 1,m is obtained by attaching K 1,m to each vertex of P n . Theorem 1.8 [5]: Any Path is a Geometric mean graph.

Theorem 1.9 [5]: Any Cycle is a Geometric mean graph.

Theorem 1.10 [5]: Combs are Geometric mean graphs.

Theorem 1.11 [5]: Crowns are Geometric mean graphs.

2. MAIN RESULTS

Theorem 2.1: Let G be the graph obtained by attaching K 1,2 at each vertex of P n . Let G 1

be the graph obtained by subdividing the path P n . Then G 1 is a Geometric mean graph.

Proof: Let G be the given graph with the path v ₁ , v ₂ ….v _n

Let u i and w i be the vertices of K 1,2 which are attached to each vertex v i of the Path P n . Let G 1 be the graph which is obtained by subdividing the edges of P n .

Let t ₁ , t ₂ ….t _n-1 be the vertices which subdivide the path P _n .

Then the graph G 1 contains 4n-1 vertices and 4n-3 edges and the graph G 1 is given

below.

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Figure: 1 Define a function f: V(G 1 ) → {1,2….q+1} by f(v 1 ) = 2

f(v i ) = 4i-3, 2≤i≤n

f(u 1 ) = 1, f(u i ) = 4i-2, 2≤i≤n f(w 1 ) = 3, f(w i ) = 4i-1, 2≤i≤n f(t i ) = 4i, 1≤i≤n

Edges are labeled with f(v i u i ) = 4i-3, 1≤i≤n f(v i w i ) = 4i-2, 1≤i≤n f(w i t i ) = 4i-1, 1≤i≤n

f(t i v i+1 ) = 4i , 1≤i≤n

From the above labeling pattern, we get distinct edge labels.

Hence f provide a Geometric mean labeling of G.

Example 2.2: Subdividing the path P 4 in (P 4 .K 1,2 ) is shown below.

Figure: 2

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Theorem 2.3: Subdividing the Path P n in (P n .K 1,3 ) is a Geometric mean graph

Proof: Let G be the given graph obtained by attaching K _1,3 at each vertex of P _n and it is denoted by P n .K 1,3 Let u i , w i , t i be the vertices of K 1,3 which are attached to vertex v i of the path P n . Let G 1 be the graph obtained by subdividing the edges of P n and t 1 , t 2 ….t n-1

be the vertices which subdivide the path P _n .

Then the graph G ₁ contains 5n-1 vertices 5n-2 edges and the graph G ₁ is shown below.

Figure: 3 Define a function f: V(G) → {1,2….q+1} by f(v i ) = 5i-3, 1≤i≤n

f(u i ) = 5i-4, 1≤i≤n f(w i ) = 5i-2, 1≤i≤n f(x i ) = 5i-2, 1≤i≤n f(t i ) = 5i, 1≤i≤n Edges are labeled with f(u i v i ) = 5i-4, 1≤i≤n f(v i w i ) = 5i-3, 1≤i≤n f(v i x i ) = 5i-2, 1≤i≤n f(v i t i ) = 5i-1, 1≤i≤n f(t _i v _i+1 ) = 5i, 1≤i≤n

Hence f provide a Geometric mean labeling of G.

Example 2.4: Subdividing the path P ₄ of (P ₄ . K _1,3 ) is a Geometric graph and it is shown

below.

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Figure: 4

Theorem 2.5: Let G be the graph obtained by attaching pendant edges to both sides of each vertex of a path P _n . Let G ₁ be the graph obtained by subdividing the path P _n then G ₁ is a Geometric mean graph.

Proof: Let G be the given graph obtained by attaching pendant edges to both sides of each vertex of a path P _n . Let u _i , v _i , w _i be the vertices which subdivide the path P _n and graph G ₁ is given below.

Figure: 5 Define a function f: V(G) →{1,2….q+1} by f(v _i ) = 4i-2, 1≤i≤n

f(u _i ) = 4i-1, 1≤i≤n

f(w _i ) = 4i-3, 1≤i≤n

f(t _i ) = 4i, 1≤i≤n-1

Edges are labeled with

f(v i u i ) = 4i-2, 1≤i≤n

f(u _i w _i ) = 4i-3, 1≤i≤n

f(v i t i ) = 4i-1,1≤i≤n

f(t _i v _i+1 ) = 4i,1≤i≤n

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Thus f provides a Geometric mean labeling of G.

Example 2.5: The labeling pattern is shown in the following figure.

Figure: 6

Theorem 2.6: Subdividing the path P _n in (P n AK 1 ) is a Geometric mean graph.

Proof: Let G be a given graph. Let P _n be the path u ₁ u ₂ ……u _n and v _i be the pendant edges attach to each vertex of P n .

Let G 1 be the graph obtained by subdividing the edges of G.

Here we consider the following cases.

Case (i): Let G 1 be the graph obtained by subdividing each edge of the path P n and t 1 t 2 …..t n-1 be the vertices which subdivide the edges of P n .

Define a function f: V(G 1 ) → {1,2….q+1} by f(u i ) = 3i-2, 1≤i≤n

f(v i ) = 3i-1, 1≤i≤n f(t i ) = 3i, 1≤i≤n-1 Edges are labeled with f(u i v i ) = 3i-2, 1≤i≤n f(u i t i ) = 3i-1, 1≤i≤n f(t i u i+1 ) = 3i, 1≤i≤n

Hence G ₁ is a Geometric mean graph and it is shown below.

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Figure: 7

Case (ii): Subdividing each pendant edges u i v i in the comb Let w _i be the vertices which subdivide the edges u _i and v _i Define a function f : V(G ₁ ) → {1,2,…,q+1} by

f(u _i ) = 3i, 1≤i≤n f(v _i ) = 3i-2, 1≤i≤n f(w _i ) = 3i-1, 1≤i≤n Edges are labeled with f(u _i w _i ) = 3i-2, 1≤i≤n f(w _i v _i ) = 3i-1, 1≤i≤n f(u _i u _i+1 ) = 3i, 1≤i≤n

Thus f provide a Geometric mean labeling for G and it is shown below.

Figure: 8 Case (iii): Subdividing each edge of a comb G.

Let t i be the vertices which subdivide the edges u i and u i+1 and w i be the vertices which subdivide the edges u _i and v _i .

Then define a function

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f: V(G 1 ) → {1,2….,q+1} by f(u i ) = 4i-2, 1≤i≤n

f(t _i ) = 4i, 1≤i≤n-1 f(w _i ) = 4i-1, 1≤i≤n f(v _i ) = 4i-3, 1≤i≤n Edges are labeled with f(u i t i ) = 4i-1, 1≤i≤n f(t i u i+1 ) = 4i, 1≤i≤n f(u i w i ) = 4i-2, 1≤i≤n f(w _i v _i ) = 4i-3, 1≤i≤n

Thus f provide a Geometric mean labeling for G 1 .

From case (i), (ii), (iii), we conclude that G ₁ is a Geometric mean graph.

Theorem 2.6: Let G be a comb. G ¹ be the graph obtained by attaching K 1,2 at each pendent vertex of comb. Let G 1 be the graph obtained by subdividing each edge of a path in G ¹ . Then G ¹ is a Geometric mean graph.

Proof: Let G be a comb and G ¹ be the graph obtained by attaching K 1,2 at each pendent vertex of G.

Let its vertices be u i ,v i , x i and y i (1≤i≤n) Let G 1 be the graph obtained by subdividing each edge of a path in G ¹ and the graph is shown below.

subdividing each edge of a path in G ¹ and the graph is shown below.

Figure: 9 Define a function f: V(G ₁ ) → {1,2,….,q+1} by f(u _i ) = 5i-4, 1≤i≤n

f(v _i ) = 5i-2, 1≤i≤n

f(x _i ) = 5i-3, 1≤i≤n

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f(y i ) = 5i-1, 1≤i≤n f(t _i ) = 5i, 1≤i≤n-1

From the above labeling pattern, we get distinct edge labels.

Hence G′ is a Geometric mean graph

Example 2.7: The labeling pattern is shown in the following figure.

Figure: 10

Theorem 2.8: A graph obtained by attaching a triangle at each pendent vertex of a comb G. Let G ₁ be the graph obtained by subdividing each edge of a path P _n of G. Then G ₁ is a Geometric mean graph.

Proof : Let G be a comb and the graph is obtained by attaching a triangle at each pendent vertex of a combits and vertices be u i , v i , x i and y i (1≤i≤n)

Let G ₁ be the graph which is obtained by subdividing each edge of a path P _n of G and t ₁ t ₂ ….t _n-1 be the vertices which subdivide the path u _i and u _i+1

The labeling pattern is shown in the following figure

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Figure : 11 Then define a function f: V(G ₁ ) → {1,2….q+1} by f(u i ) = 6i-5, 1≤i≤n

f(v i ) = 6i-3, 1≤i≤n f(x i ) = 6i-2, 1≤i≤n f(y i ) = 6i-1, 1≤i≤n f(t i ) = 6i, 1≤i≤n-1

From the above labeling pattern we get distinct edge labels.

Hence f is a Geometric mean labeling for G ₁ and G ₁ is a Geometric mean graph.

Example 2.9: The labeling pattern shown in the following figure.

Figure: 12

Theorem 2.10: Let G = P _n ∆K 3 is a Geometric mean graph. Let G 1 be the graph which is

obtained by subdividing the path P _n of G. Then G ₁ is a Geometric mean graph.

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Proof: Let G be a given graph its vertices be x i , y i and u i (1≤i≤n) Let G 1 be the graph obtained by subdividing the path of G.

Let t _i be the new vertices which subdivide the path of u _i and u _i+1 Then define a function f: V(G) →{1,2….q+1}

by f(u _i ) = 5i-4, 1≤i≤n f(x 1 ) = 5i-4, 1≤i ≤ _n f(y i ) = 5i-1, 1≤i ≤ _n f(t i ) = 5i, 1≤i ≤ _n-1

From the above labeling pattern, we get distinct edge labels.

Hence f is a Geometric mean labeling of G ₁ .

The labeling pattern is shown in the following figure.

Figure: 13

Theorem 2.11: S(L _n ) is a geometric mean graph

Proof: Let L n denote the ladder graph which is has 2n vertices and 3n-2 edges and its vertices be u _i and v _i (1≤i ≤ _n)

Let L N be the graph obtained by subdividing all the edges of L n and its vertices be x i , y i

and w i

The labeling pattern of a graph L _N is given below.

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Figure: 14

Then define a function f: V(L _N ) → {1,2….,q+1}by f(u _i ) = 6i-3, 1≤i≤n

f(v _i ) = 6i-5, 1≤i≤n

f(w i ) = 6i-4, 1≤i≤n-1

f(y i ) = 6i-2, 1≤i≤n-1

f(x i ) = 6i-1, 1≤i≤n-1

Edges are labeled with

f(u _i w _i ) = 6i-3, 1≤i≤n

f(w i v i ) = 6i-5, 1≤i≤n

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f(u i x i ) = 6i-2, 1≤i≤n f(x _i , u _i+1 ) = 6i 1≤i≤n f(v i y i ) = 6i-4 1≤i≤n f(y _i v _i+1 ) 6i-1, 1≤i≤n

Hence f provide a Geometric mean labeling for S(L n ).

Hence S(L n ) is a Geometric mean graph.

Example 2.12: The labeling pattern of S(L _n ) is shown in the following figure.

Figure 15

REFERENCES

1. Gallian, J.A.(2010). A Dynamic survey of Graph labeling. The Electronic Journal of Combinatories 17 (#DS6) (2010).

2. Frank Harary, Graph Theory, Narosa publishing House Reading New Delhi (2001)

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3. Somasundaram,S. and Ponraj,R.(2003). Mean Labeling of Graphs, National Academy of Science letters, Vol 26 (2003), p210-213.

4. Somasundaram, S., Ponraj,R. and Sandhya,S.S.(2014). Harmonic Mean Labeling of Graphs Communicated to Journal of Combinatorial Mathematics and Combinatorial Computing.

5. Somasundaram,S., Vidhyarani,P. and Ponraj,R.(2011). Geometric mean labeling of graphs,

Bulletin of Pure and Applied Sciences 30E(2) (2011), p153-160.