Square Difference Labeling of Theta Graphs

Square Difference Labeling of Theta Graphs

Department of Mathematics

Abstract:- In this work, we investigate Square Difference labeling of theta graph (T), We also discuss SDL in some graph operations namely Fusion, Duplication, Switching, path union and one point union of r copies of T graph.

Keyword: Square difference graph (SDG), Square difference labeling (SDL), fusion, duplication, switching, P.r(T), one point union.

AMS Classification: 0578

1. INTRODUCTION

   Graph labeling was first introduced in the mid sixties. A dynamic survey on graph labeling is regularly updated in [4]. For all other terminology and notations in graph theory follow [3]. The square Sum labeling is previously defined by [1]. The concept of SDL was first introduced by [6]. CDL for some special graphs and some graphs is proved [5, 2].

2. DEFINITIONS

   Definition 2.1

   A graph G = (p, q) with x vertices and y edges is said to admits a square difference labeling, if there exist a bijection f: V {0,1,2, 1} such that the induced function f*: E N given by f*(ab) = |[()]2 [()]2| is injective, . A graph which admits square difference labeling is called square difference graph.

   Definition 2.6

   Let G = G1 = G2 = … = Gn be graph for n 2. Then the graph is reduced by adding an edge from Gi to Gi+1 for i=1 to n-1 is called the path union of G.

3. MAIN RESULTS

Theorem 3.1

The Theta graph (T) is a SDG.

Proof:

Let T, with centerv0and the edge set be

E(G) = {+1/1 5} {01, 04} {16}

Now, |(T)| = 7 and |(T)| = 8

Define a vertex labeling as

f(vi) = i 1, 1 6, f(v0) = 6, now the edge labelings are defined as,

f *(01) = [(0)]2, (04) = |[(0)]2 [(4)]2|,(+1) = 2 1 for = 1 5 , (16) = [(6)]2

Thus, () (), () and now edge labelings are distinct. Hence the graph Tadmits SDL. For instance, T6is given below.

Definition 2.2

A theta graph (T) is a segment with two different vertices of degree 3 and all other vertices of degree 2.

Definition 2.3

A vertex is said to be a duplication of if all the vertices

Theorem 3.2

Fig 3.1 T6 admits SDL

which are adjacent to

Definition 2.4

are now adjacent to

.

The duplication of any vertex vi of degree 3 in the cycle of T

admits square difference graph.

Proof:

Let G be a (p, q) graph. Let u v be two vertices of G. We replace them with single vertex w and all

Edges incident with u and that with v are made incident with

w. If a loop formed is deleted, the new graph has p-1 vertices and q-1 edges are called fusion of graphs [3].

Definition 2.5

A vertex switching Gv of a graph is obtained by taking a vertex c of G, detach all the edges incident with c and connect

edges joining c to every vertex which are not adjacent to c in

Let G be the graph obtained by duplication of any vertex vi in T and be the duplicating vertex of vi of degree 3. In T only two vertices are of degree 3. i.e.,v1 and v4.

Consider, V(G) = {/0 7}and E(G) = {+1/1 5} {01 , 04, 16}.

Clearly, |()| = 8 and |()| = 11

Now, define a vertex label : {0,1, 7} as

f(vj) = j 1, 1 6

f(v0) = 6

( ) = 7, where is the duplicating vertex of v1. For the

1 1

1. above pattern the edges are defined as same as in theorem3.1 and are distinct similarly,

   Case (i) : Duplication of v1 , the edges are labeled as follows

   (01 ) = 13, (21 ) = 48, (61) = 24. Thus, all the

   1 1 1

   edge labeling are distinct and are illustrated in figure 3.1(a)

   Example 3.1

   Fig 3.3 The path union of 2(T6) is SDL

   Theorem 3.4.

   The one point union of r copies of theta graph admits SDL. Proof:

   Consider G = (V, E) be the one point union of r(T). Now define V(G) = {() , /0 5,1 } and

   E(G) = {()() } {()} {()}

   Fig 3.1(a) Duplication of v1 in T6 admits CDL

   {()()} {()}

   +1 5

   Case (ii) : Duplication of v4, the edges are labeled distinct as 0 3 0

   ( 1 ) = 13, ( 1 ) = 45, ( 1) = 33 and an

   Clearly, |()|= 6r+1 and |()|= 8r

   0 4 3

   ex is illustrated

   4 5 4

   igure 3.1(b).

   Define the vertex function as

   ample

   in f

   (()

   0

   0

   = 6r

   ) = 6j-1, (()

   ) = i+6j-7 for 1 5, 1 , f(w)

   Fig 3.1(b) Duplication of v4 in T6 admits SDL

   The edge set E is classified as same as mentioned in the theorem 3.3. Thus() (), (). Hence the theorem is verified. For instance, the example of 4(T) given below.

   Theorem 3.3

   The P.r(T) is a SDG.

   Proof:

   Assume the graph r(T) with the vertex set V = (), 0 5,

   1 and the edge set E(G) = E1 2 3, where

   E1 = {()() /1 5}

   +1

   E2 = {()(), ()()/1 }

   0 1 0 4

   E3 = {()(+1)/1 1} and || =7r and || =9r-1.

   3 2

   0

   0

   Vertex function is defined as h: {0,1, 7 1} (()) = 7j-1

   (()) = i+7j-8

   The edge set E is specified as E1 and E2.

   E1: The edges formed have an increasing sequence of even integers, if the end of the two vertices have either odd or even.

   E2: The edges formed have an increasing sequence of odd integers, if one end of vertex has odd integer and other has even integer.

   Hence the condition is satisfied. Therefore the graph T is square difference graph. The example for the above graph is shown below.

   Fig 3.4. SDL of One point union of 4(T6)

   Theorem 3.5

   The fusion of any two vertices in the cycle of T is SDL. Proof:

   Let T be the graph with centre v0, V = {u0, u1,u6} and E =

   E1 2 , where E1 = {+1/1 6} and E2 =

   {01, 04, 16}

   Also the cardiality of vertices and edges is noted as 7 and 8 respectively.

   Now, we obtain a graph G by fusing two vertices5 and 6 in

   T and we name it as 5. After the fusion

   |()| = 6 and |()| = 7. Define the function :

   {0,1, 5} as

   f(u0) = 5

   f(ui) = i-1

   Then the edge yields the labeling as,

   f *( 01) = [(0)]2 , (04) = |[(0)]2 [(4)]2| ,

   (+1) = 2 1 for i = 1 to 4, (15) = [(5)]2.

   It is easily observed that all the edge labels are distinct. Hence the graph G admits SDL. For the above graph, the example

   mentioned in fig 3.5

   Fig 3.5 (a) Theta graph

   Fig 3.5 (b) the fusion of u5 and u6 in theta graphs is SDG

   Theorem 3.6

   The switching of a central vertex in T is square difference graph.

   Proof

   Let the graph G is obtained by switching the central vertex x0in T with the vertices0,1, , 6 and the edges {+1/ 1 5} {02, 03, 05,06} {16}

   The cardinality of vertices and edges are 7 and 10 resp.,

   Consider the one to one function : {0,1, 6} as

   f(x0) = 6

   f(xj) = j-1

   We label the edge as

   (+1) = 2 1, 1 5, (16) = 25, (02) = 35, (03) = 32, (05) = 20,

   (06) = 11.

   <>Clearly, the entire 10 edge labels are distinct. Therefore, the graph G is square difference graph.

   Example 3.2

   Fig 3.6 The switching of x0 in theta graph is SDG

   CONCLUSION:

   From this work we conclude that T and its associated graphs are Square difference graph.

   REFERENCES

   1. Ajitha.V, Arumugam.S and Germina.K.A, On square Sum graphs, AKCE J. Graph, combin, 6(2006) 1-10.

   2. Arthy. J Cube Difference Labeling of Some Graphs, AIP Conference Proceedings 2112, 020147 (2019).

   3. B West, Introduction to graph theory, Prentice Hall, India, 2001.

   4. Gallian. J.A. A dynamic survey of graph labeling, The Electronics journal of Combinatories, 17(2010) #DS6.

   5. P. Jagadeeswari, Cube Difference labeling of Theta Graphs, International journal of Innovative Science and Research Technology, Vol 4, issue 5, May 2019.

   6. Shiama. J, Some special types of Square difference graphs , Journal of Mathematical Archives, Vol.3(6), 2012, 2369-2374, ISSN 2229-5046.