 Open Access
 Authors : K. Bhuvaneswari , P. Jagadeeswari
 Paper ID : IJERTV8IS100212
 Volume & Issue : Volume 08, Issue 10 (October 2019)
 Published (First Online): 30102019
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
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

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 n1}
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.

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 6n2 respectively. Now, define the function f as
f(xi,) =2(i 1),
f(yi,) = 2i 1
f(xi,j) = 2n +4i+2j5 f(yi,j) = 2n +4i+2j6
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

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 (k1) 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) ) = 4n21 + (8n24n)(k1)
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 8nr1 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(k1)
*(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(k1)
C(r, Hn
2)) is SDG.
i i 1
Proof
f *( x(k ) y(k ) ) = 2n 1+4n(k1) 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(k1)
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(k1)
i i 1
increasing. Hence, the theorem is proved.
f *( g (k ) l ( k ) ) = 2n 1+4n(k1) 1(mod 4)( n is odd)
n1 n 1
2 2
f *( g (k ) l ( k ) ) = 2n 1 + 4n (k1) 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 n1; 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 — p1} 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.

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

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