Square Difference for Some Path Union and Duplication of Graphs

Square Difference for Some Path Union and Duplication of Graphs

K. Bhuvaneswari 1

1Department of Mathematics, Mother Teresa Womens University, Kodaikanal,

Tamil Nadu, India

P. Jagadeeswari 2

2Department of Mathematics, BIHER, Chennai, Tamil Nadu, India

(1Ph.D Scholar, MTWU, Kodaikanal)

Abstract:- In this paper, we prove that cycle union of r copies of HnK2, open star of r copies of HnK2, corona of Hn with

, path union of corona of Hn with are Square Difference graph (SDG).

Theorem 2.1.1

The graph Hn

Proof:

Consider the graph Hn

2(n 3) is a SD graph.

2 with

Keywords:- Square difference graph, duplication, path union, cycle union, open star

AMS classification : 05C78

1. INTRODUCTION

   Throughout this work, we use finite, undirected, simple

   V = {xi, yi, xi,j, yi,,j /1 i n; 1 j 2} E = E1 E2 E3 where,

   E1 = {xi, xi+1, yi yi+1 /1 i n-1}

   xn1 yn1 , n odd

   2 2

   E2 =

   graph and we follow [1,6]. In [4, 5] proved some pyramid

   xn

   yn , n even

   graphs and H – graphs for square difference. Square sum labelling for pyramid graph, Square Difference labeling of theta graphs and PCL of corona of HnK2 are proved by Subashini et. al. [7, 8, 9]. Prime Cordial Labeling of H- graph and its related graphs are established in [9]. Thousands of labeling are surveyed and revised be Gallian [3]. Cube difference labelling for H graph were proved by [2]. Motivated be their work, in this paper we prove the union and duplication of some graphs.

2. MAIN RESULTS

2 1 2

E3 = {xixi,j , yiyi,j /1 i n; 1 j 2}

Clearly, the cardinality of the vertices and edges are 6n and 6n-2 respectively. Now, define the function f as

f(xi,) =2(i 1),

f(yi,) = 2i 1

f(xi,j) = 2n +4i+2j-5 f(yi,j) = 2n +4i+2j-6

and we receive the edge labels f * as follows:

f *(xi xi+1) = 8i 4, 1 i n f *(yi yi+1) = 8i, 1 i n

1. Union and open star of corona graphs

   In my previous work, I proved that Hn graph (n 3), P(r,

   f *(x n1

   2

   f *( x

   yn1 ) = 2n 1 1(mod 4)

   2

   y ) = 2n 1 3(mod 4)

   Hn), C(r, Hn), S(r, Hn), HnK1 etc., [5]. By continuing that, in this paper we prove some Hn related graphs for SDG. For definition of path union, cycle union and open star refer [5].

   Definition 2.1.1.[4]

   A graph G = (p, q) is said to be a square difference graph if

   n 1 n

   2 2

   Thus, the entire 6n 2 edges are distinct. Hence the theorem.

   Example 2.1.1.

   it admits a bijective function g: V {0, 1, 2, p1}

   such that the induced function g *: E(G) N given by g*(xy)

   = |[g(x)]2 [g(y)]2| are all distinct, xy E(G).[6].

   Square difference labeling for H3

   2and H4

   2.

   Definition 2.1. 2.[9]

   An Hn (n 3) graph is obtained by the two paths

   P1 and

   n

   n

   n

   n

   P2 with the vertices u1, u2,un and v1,v2,vn respectively

   and joining the vertices un1

   2

   +1 n

   +1 n

   odd otherwise and v .

   2

   2

   and vn1 by an edge, if n is

   2

   Figure 2.1. SDL for H32and H42

   Theorem 2.1.2.

   The path union of H (n 3) admits SDL.

   Figure 2.2(a). SDL for P(H32)

   f *( u (k ) v(k ) ) = 2n 1 + 4n (k-1) 3(mod 4)

   n 2

   Proof:

   n 1 n

   2 2

   Let H (n 3) be the corona graph of H

   with

   with

   f*( u(k )u(k ) ) = f *( v(r ) v(r ) ) 16n(k 1) 16i,1 i n

   n 2 n 2

   i,1

   i,2

   n1 n

   the vertex set,

   f *( v(k ) v(k ) ) = f *( v(r ) v(r ) ) 16n(k 1) 16i 4,1 i n

   V = V1 V2, where,

   i,1

   i,2

   n1 n

   ( k )

   f *(v ( k ) u (k 1) ) = 4n2-1 + (8n2-4n)(k-1)

   i

   i

   V1 = {x i , y ( k ) / 1 i n, 1 k r} and 1 1

   V2 = { x(k ) , y(k ) /1 i n,1 j 2;1 k r}

   i, j i, j

   and the edges E = 4=1 , where,

   E1 = x(k ) x(k ) , y(k ) y(k ) /1 i n 1;1 k r;

   i i1 i i1

   x(k ) y ( k ) , n odd

   E2 =

   n1 2

   n1

   2

   x(k ) y ( k ) , n even,1 k r

   n 1 n

   2 2

   E3 = x(k ) x(k ) , y(k ) y(k ) /1 i n;1 j 2;1 k r

   i i, j i i, j

   E4 = y(k ) x(k 1) /1 k r 1

   1 1

   It is obvious that, the number of vertices and edges

   are 6nr and 8nr-1 resp.,

   Also, define the vertex labeling function as follows: For 1 i n, 1 j 2, 1 k r,

   i

   i

   f (x(k ) ) 2(i 1) 2n(k 1)

   i

   i

   f ( y(k ) ) 2i 1 2n(k 1)

   f (x(k ) ) f ( y(r ) ) 4i 2 j 4 4n(k 1)

   Figure 2.2(b). SDL for P(H42)

   i, j n

   f ( y (k ) ) f ( y (r ) ) 4i 2 j 5 4n(k 1)

   i, j n

   Thus, all the edge labeling are distinct i.e., f *(e ) f

   Thus, the induced function f *: E(Hn2) N satisfies i

   the condition of SD labeling and the edges of Hn2

   receives label as, For 1 k r,

   f *(x ( k ) x ( k ) ) = 8i 4 + 8n(k-1)

   *(ej), ei ej E(G) . Hence P(r, Hn

   graph admits Square difference labeling.

   Theorem 2.1.3.

   2), (n 3)

   i i 1

   f *( y ( k ) y ( k ) ) = 8i + 8n(k-1)

   C(r, Hn

   2)) is SDG.

   i i 1

   Proof

   f *( x(k ) y(k ) ) = 2n 1+4n(k-1) 1(mod 4)( n is odd)

   Consider, G = C(r, HnK2) be the graph.

   n 1 2

   n 1 2

   Let V = V V , where,

   1 2

   V1 = {g ( k ) , l ( k ) / 1 i n, 1 k r} and

   i i

   V2 = { g (k ) ,l (k ) /1 i n,1 j 2;1 k r}

   i, j i, j

   and the edges E = 6=1 , where,

   E1 = g (k ) g (k ) ,l (k )l (k ) /1 i n 1;1 k r ;

   i i1 i i1

   g ( k ) l ( k ) , n odd

   E2 =

   n1 n1

   2 2

   g ( k ) l ( k ) , n even,1 k r

   n 1 n

   2 2

   E3 = g (k ) g (k ) ,l (k )l (k ) /1 i n;1 j 2;1 k r

   i i, j i i, j

   E4 = g (k ) g (k ) ,l (k )l (k ) /1 i n;1 k r

   i,1 i,2 i,1 i,2

   E5 = l (k ) g (k 1) /1 k r 1

   1 1

   E6 = {()(1)}

   1

   Clearly,

   1

   V

   E(G)

   = 6nr and

   = 8nr

   Also, we receive vertex and edge labeling as For 1 i n; 1 j 2; 1 k r

   i

   i

   f (g (k ) ) 2(i 1) 2n(k 1)

   i

   i

   f (l (k ) ) 2i 1 2n(k 1)

   f (g (k ) ) f (l (r ) ) 4i 2 j 4 4n(k 1)

   i, j n

   i, j n

   i, j n

   f (g (k ) ) f (l (r ) ) 4i 2 j 5 4n(k 1)

   Figure 2.3(b) SDL of C(3, H3K2)

   Theorem 2.1.4.

   The graph S(r, HnK2) admits SDL.

   Proof:

   Let G = S(r, HnK2) with V = V1 V2, where,

   V1 = {g ( k ) , l ( k ) / 1 i n, 1 k r} and

   i i

   V2 = { g (k ) ,l (k ) /1 i n,1 j 2;1 k r}

   i, j i, j

   And E = 5=1 , where,

   E1 = g (k ) g (k ) ,l (k )l (k ) /1 i n 1;1 k r ;

   i i1 i i1

   g ( k ) l ( k ) , n odd

   E =

   n1 n1

   2 2 2

   g ( k ) l ( k ) , n even,1 k r

   n 1 n

   2 2

   E3 = g (k ) g (k ) ,l (k )l (k ) /1 i n;1 j 2;1 k r

   i i, j i i, j

   E4 = g(k ) g (k ) ,l (k )l (k ) /1 i n;1 k r

   i,1 i,2 i,1 i,2

   Figure 2.3(a) SDL of C(3, H4K2)

   Using this induced function f *, the edges of G receives labeling as same as mentioned in theorem 3.4.4 and added to

   E5 = wl(k ) /1 k r

   1

   1

   We know that, the cardinality of vertices and edges are 6nr

   +1 and 8nr resp.,

   And the vertex valued function are as same as mentioned in the above theorem and added to f(w) = f(()) + 1.

   The induced function f * receives the edge labels as

   * () (1)

   () 2

   f *(g ( k ) g ( k ) ) = 8i 4 + 8n(k-1)

   f (1 1 ) = [f((1 )] – 1

   i i 1

   Hence, all the edge labeling are distinct and strictly

   f *( l ( k ) l ( k ) ) = 8i + 8n(k-1)

   i i 1

   increasing. Hence, the theorem is proved.

   f *( g (k ) l ( k ) ) = 2n 1+4n(k-1) 1(mod 4)( n is odd)

   n1 n 1

   2 2

   f *( g (k ) l ( k ) ) = 2n 1 + 4n (k-1) 3(mod 4)

   n 1 n

   2 2

   f *( g (k ) g (k ) ) = f *( l (r ) l (r ) ) 16n(k 1) 16i,1 i n

   i,1

   i,2

   n1 n

   f *( l (k )l (k ) ) = f *( l (r ) l (r ) ) 16n(k 1) 16i 4,1 i n

   i,1

   i,2

   n1 n

   1

   1

   f *(w()) = 0(mod 2)

   Figure 2.5. Duplication of vertex by a vertex of C3

   Theorem 2.2.1.

   The Duplication of any pendant vertex of pyramid graph Jn (n 3), is SDG.

   Proof:

   Let G be the graph obtained by duplication of any

   Figure 2.4(a) SDL of S(3, H3K2)

   i i, j i i, j

   i i, j i i, j

   and f *( g (k ) g (k ) ,l (k )l (k ) ) is in the form of an increasing

   order of odd integer when its one end vertex is odd integer and the other end vertex is even integer.

   pendant vertex of Jn. , be the duplication vertex of , of degree one. In Jn, only two vertices are pendant vertices. i.e. ,1, and ,.

   Consider,

   V(G) = {, / 1 i n; 1 j i} { ,1, ,} and

   the edge set

   E(G) = {,+1,, ,+1,+1 / 1 i n-1; 1 j i}

   { , , }

   ,1 1,1 , 1,1

   Then,

   V

   E(G)

   = ( +) + 2 = p and

   2

   2

   2

   = (2 ) + 2 = q

   Let the function f : V {0,1, 2 — p-1} defined as follows:

   f( ui,j) =

   1 i(i 1) ( j 1) for 1 i n ; 1j i

   2

   f( ,1) = f(,) + 1

   ,

   ,

   f( ) = f (

   ,

   ) + 2

   Figure 2.4(b) SDL of S(3, H4K2)

   Clearly, the above given labeling satisfies the condition of SDL and receives the distinct edge labeling.

   Hence the theorem is verified.

   i

   i

   From the above, f *(ei) f *(ej), e

   ej

   E(G) . Hence

   Example 3.1.2.

   The Duplication of pendant vertex of J5 is SDG.

   S(r, HnK2), (n 3) graph admits Square difference labeling.

2. Duplication of a pendant vertex of pyramid and hanging pyramid graph

   Definition 2.2.1.

   A vertex is said to be a duplication of if all the vertices which are adjacent to are now adjacent to .

   Example 2.2.1.

   Duplication of vertex by a vertex of C3.

   Figure 2.6. SDL for duplication of pendant vertex of J5.

   Theorem 2.2.2.

   The duplication of any pendant vertex of hanging pyramid graph admits SDL.

   Proof:

   Consider the graph G procured by duplicating the pendant vertex of hanging pyramid graph.

   In HJn, the vertices u0 , un,1 , un,n are the pendant

   vertices.

   The vertex set and edge set are same as theorem 2.4.1. added to { } and { } respectively.

   CONCLUSION

   In this work, we investigated that the path union, cycle union, open star and duplication of some graphs admits Square difference labelling.

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      Example 2.4.2

      The duplication of pendant vertex of HJ4 is SDL.

      Figure 2.7. SD for duplication of pendant vertex of hanging pyramid graph.

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