Square Difference for Some Path Union and Duplication of Graphs

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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.

      REFERENCES

      1. Frank Harary, Graph theory (Narosa Publishing House, 2001).

      2. J. Arthy, K. Manimekalai and K. Ramanathan, Cube Difference Labeling of some special graphs, AIP Conference Proceeding 2112, 020147(2019) http://doi.org/10.1063/1.511232.

      3. J.A. Gallian, A dynamic survey of graph labeling, The Electronics

        0

        Then,

        0

        V

        E(G)

        0

        2

        2

        = ( +) + 3 and

        2

        = (2 ) + 3

        journal of Combinatories, 17(2010) #DS6.

      4. P. Jagadeeswari, K. Ramanathan and K. Manimekalai, Square Difference Labeling for pyramid graph and its related graphs, International journal of Mathematics And its Applications, 6(1), 9196 (2018).

      5. P. Jagadeeswari, K. Manimekalai and K. Ramanathan, Square

        Let the vertex valued function f are defined as,

        f(u0) = 0

        Difference Labeling for H-graphs, AIP Conference Proceeding 2112, 020147(2019) http://doi.org/10.1063/1.511232.

      6. A. Rosa, On certain valuation of graph, theory of graphs (Rome,

        f(u

        i,j) =

        1 (i 2 i) j, 1 i n; 1 j i

        2

        july 1966), Golden and Breach. N.Y and Paris, 349355 (1967).

      7. G.Subashini, K. Bhuvaneswari, K. Manimekalai, Square Difference labeling of Theta Graphs, International Journal Of

        0

        0

        f( ) = f (un,n) + 3

        Then, the induced edge function f * for the above labelling

        Engineering And Research Vol-8 , issue 9.

      8. G. Subashini, K. Ramanathan and K. Manimekalai, Square sum

        pattern are distinct. i.e., f

        *(ei)

        f *(ej) e e E(G)

        i j .

        Labeling for Pyramid Graph and Hanging Pyramid Graph, International journal of Mathematics and its application, 6(1), 97- 102, 2018.

        Therefore, the duplication of pendant vertex of HJn admits

        SDL.

        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.

      9. G. Subashini, K. Manimekalai and K. Ramanathan, Prime Cordial Labeling for HnK2, AIP Conference Proceeding 2112, 020147(2019) http://doi.org/10.1063/1.511232.

      10. Sugumaran and Mohan, Prime Cordial labeling of some graphs related to H graph, Annals of Pure and Applied Mathematics, 16(1), 171183 (2018).

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