2. Super Pair Sum Graphs

Volume 3, Issue 2 (2015), 177–183.

ISSN: 2347-1557

Available Online: http://ijmaa.in/

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International Journal of Mathematics And its Applications

Super Pair Sum Labeling of H-graphs

Research Article

J.Venkateswari¹, R.Vasuki^2∗ and P.Sugirtha³

1 Department of Mathematics, Dr. Sivanthi Aditanar College of Engineering, Tiruchendur-628 215, Tamil Nadu, India.

2 Department of Mathematics, Dr. Sivanthi Aditanar College of Engineering, Tiruchendur-628 215, Tamil Nadu, India.

3 Department of Mathematics, Dr. Sivanthi Aditanar College of Engineering, Tiruchendur-628 215, Tamil Nadu, India.

Abstract: Let G be a graph with p vertices and q edges.The graph G is said to be a super pair sum labeling if there exists a bijection ffrom V (G) ∪ E(G) ton

0, ±1, ±2, . . . , ±

p+q−1 2

o

when p + q is odd and from V (G) ∪ E(G) ton

±1, ±2, . . . , ±

p+q 2

o when p + q is even such that f (uv) = f (u) + f (v). A graph that admits a super pair sum labeling is called a super pair sum graph. In this paper, we investigate super pair sum labeling of some H-graphs.

MSC: 05C78.

Keywords: labeling, super pair sum labeling, super pair sum graph.

c

JS Publication.

1. Introduction

Throughout this paper, by a graph we mean a finite, undirected and simple graph. Let G(V, E) be a graph with p vertices and q edges. For notations and terminology we follow [2]. Path on n vertices is denoted by Pn and a cycle on n vertices is denoted by Cn. K1,mis called a star and it is denoted by Sm. The bistar Bm,nis the graph obtained from K2 by identifying the center vertices of K1,m and K1,n at the end vertices of K2 respectively. Bm,n is often denoted by B(m). The union of two graphs G1 and G2 is a graph G1∪ G2 with V (G1∪ G2) = V (G1) ∪ V (G2) and E(G1∪ G2) = E(G1) ∪ E(G2).

The H-graph of a path Pn, denoted by Hn is the graph obtained from two copies of Pn with vertices v1, v2, . . . , vn and u1, u2, . . . , un by joining the vertices vn+1

2 and un+1

2 if n is odd and the vertices vⁿ

2+1 and uⁿ

2 if n is even. The corona of a graph G on p vertices v1, v2, . . . , vp is the graph obtained from G by adding p new vertices u1, u2, . . . , upand the new edges uivi for 1 ≤ i ≤ p. The corona of G is denoted by G  K1. The graph Pn K1 is called a comb. The 2-corona of a graph G, denoted by G  S2 is a graph obtained from G by identifying the center vertex of the star S2 at each vertex of G.

The graceful labelings of graphs was first introduced by Rosa in 1961[1] and super vertex graceful labeling of some standard graphs was discussed in [5]. The concept of pair sum labeling was introduced and studied by R. Ponraj et al. [3,4]. Let G be a (p, q) graph. A one-one map f : V (G) → {±1, ±2, . . . , ±p} is said to be a pair sum labeling if the induced edge function

∗ E-mail: [email protected]

fe: E(G) → Z − {0} defined by fe(uv) = f (u) + f (v) is one-one and fe(E(G)) is either of the form {±k1, ±k2, . . . , ±k^q

2} or {±k1, ±k2, . . . , ±kq−1

2 } according as q is even or odd. A graph with a pair sum labeling defined on it is called a pair sum graph.

Motivated by Ponraj and Parthiban, R. Vasuki et al. introduced the concept of super pair sum labeling [6] and discussed the super pair sum behaviour of some standard graphs like path, K1,m, bistar Bm,nfor m ≥ 1, n ≥ 1, P [2n; Sm] for n ≥ 1, m ≥ 1, comb, C2n, K1,m∪ K1,nand the caterpillar S(x1, x2, . . . , xn). A graph G with p vertices and q edges is said to have a super pair sum labeling if there exists a bijection f from V (G) ∪ E(G) to0, ±1, ±2, . . . , ± ^p+q−1₂
when p + q is odd and from V (G) ∪ E(G) to±1, ±2, . . . , ± ^p+q₂
when p + q is even such that f (uv) = f (u) + f (v). A graph that admits a super pair sum labeling is called a super pair sum graph.

A super pair sum labeling of P7 K1 is shown in Figure 1.

b b b b b b

b b b b b b

-1 11 -5 7 -9 3

bb

-13 13 10 -3 6 9 2 -7 -2 5 -6 -11 -10 1

12 8 4 0 -4 -8 -12

Figure 1.

In this paper, we prove that the graphs H-graph Hn, corona of a H-graph, 2-corona of a H-graph and the disconnected H-graph 2Hnfor n ≥ 3 are super pair sum graphs.

2. Super Pair Sum Graphs

Theorem 2.1. The H-graph G is a super pair sum graph.

Proof. Let u1, u2, . . . , un and v1, v2, . . . , vn be the vertices of the H-graph G. The graph G has 2n vertices and 2n − 1 edges.

Define f : V (G) ∪ E(G) → {0, ±1, ±2, . . . , ±(2n − 1)} as follows:

f (ui) =







−i, 1 ≤ i ≤ n and i is odd 2n − i + 1, 1 ≤ i ≤ n and i is even

f (vi) =



















n − i + 1, if n is odd, 1 ≤ i ≤ n and i is odd

−(n + i), if n is odd, 1 ≤ i ≤ n and i is even

−(n + i), if n is even, 1 ≤ i ≤ n and i is odd n − i + 1, if n is even, 1 ≤ i ≤ n and i is even f (uiui+1) = 2n − 2i, 1 ≤ i ≤ n − 1

f (vivi+1) = −2i, 1 ≤ i ≤ n − 1 f

 un+1

2 vn+1 2



= 0, if n is odd and f

uⁿ

2+1vⁿ

2



= 0, if n is even.

It can be verified that f is a super pair sum labeling and hence G is a super pair sum graph. For example, a super pair sum labeling of H7 and H8are shown in Figure 2.

bbbbbbb bbbbbbb bbbbbbb bbbbbbb

b b

-1 12 13

10 -3

8 11

6 -5

4 9

2 -7

0 7 -2

-9 -4

5 -6

-11 -8

3 -10

-13 -12

1 -1

14 15

12 -3

10 13

8 -5

6 11

4 -7

2 9

0 -9 -2

7 -4

-11 -6

5 -8

-13 -10

3 -12

-15 -14 H

1

H

Figure 2.

Theorem 2.2. The graph Hn K1 is a super pair sum graph.

Proof. Let u1, u2, . . . , un and v1, v2, . . . , vn be the vertices on the path of length n − 1. Let u⁰_i and v⁰_i be the pendant vertices at ui and vi respectively, for 1 ≤ i ≤ n. The graph Hn K1 has 4n vertices and 4n − 1 edges.

Define f : V (G) ∪ E(G) → {0, ±1, ±2, . . . , ±(4n − 1)} as follows:

f (u2i−1) = 4n − 4i + 3, 1 ≤ i ≤ n + 1 2



f (u2i) = 1 − 4i, 1 ≤ i ≤ n − 1 2



f (u⁰_2i−1) = 3 − 4i, 1 ≤ i ≤ n + 1 2



f (u⁰2i) = 4n − 4i + 1, 1 ≤ i ≤ n − 1 2



f (v2i−1) =







−2n − 4i + 3, 1 ≤ i ≤ⁿ⁺¹₂ and n is odd 2n − 4i + 3, 1 ≤ i ≤ⁿ₂ and n is even

f (v2i) =







2n − 4i + 1, 1 ≤ i ≤ⁿ⁻¹₂ and n is odd

−2n − 4i + 1, 1 ≤ i ≤ⁿ₂ and n is even

f (v2i−1⁰ ) =







2n − 4i + 3, 1 ≤ i ≤ⁿ⁺¹₂ and n is odd

−2n − 4i + 3, 1 ≤ i ≤ⁿ₂ and n is even

f (v2i⁰ ) =







−2n − 4i + 1, 1 ≤ i ≤ⁿ⁻¹₂ and n is odd 2n − 4i + 1, 1 ≤ i ≤ⁿ₂ and n is even f (uiui+1) = 4n − 4i, 1 ≤ i ≤ n − 1

f (uiu⁰i) = 4n − 4i + 2, 1 ≤ i ≤ n f (vivi+1) = −4i, 1 ≤ i ≤ n − 1

f (viv⁰_i) = 2 − 4i, 1 ≤ i ≤ n f

un+1 2 vn+1

2



= 0, if n is odd and f

uⁿ

2+1vⁿ

2



= 0, if n is even.

Thus, f is a super pair sum labeling and hence Hn K1is a super pair sum graph. For example, a super pair sum labeling of H7 K1 and H8 K1 are shown in Figure 3.

bbbbbbb bbbbbbb

27 24

-3 20

23 16

-7 12

19 8

-11 2 15

0 -15

-4 11

-8 -19

7 -12

-16 -23

-20 3

-24 -27 H

⊙ K

bbbbbbb

bbbbbbb

-13 17 -9 21 -5 25

-1 26 22

18

14

10

6 4

13

-17

9

-21

5

-25

1 -2

-6

-10

-14

-18

-22

-26

bbbbbbb bbbbbbb

31 28

-3 24

27 20

-7 16

23 12

-11 6 19

0 15

-4 -19

-8 11

-12 -23

-16 7

-20 -27

-24 3

H

⊙ K

bbbbbbb

bbbbbbb

-13 21 -9 25 -5 29

-1 30 26

22

18

14

10 8

-17

13

-21

9

-25

5

-29 -2

-6

-10

-14

-18

-22

-26

b b

b b

17 2 4

-15-31 -30 1 -28

Figure 3.

Theorem 2.3. The graph Hn S2 is a super pair sum graph.

Proof. Let u1, u2, . . . , unand v1, v2, . . . , vnbe the vertices on the path of length n − 1. Let uiu⁰i, uiu⁰⁰i be the path attached at ui and viv⁰i, vivi⁰⁰be the path attached at vi, 1 ≤ i ≤ n. The graph Hn S2 has 6n vertices and 6n − 1 edges. Define

f : V (G) ∪ E(G) → {0, ±1, ±2, . . . , ±(6n − 1)} as follows:

f (u2i−1) = 6n − 6i + 5, 1 ≤ i ≤ n + 1 2



f (u2i) = 1 − 6i, 1 ≤ i ≤ n − 1 2



f (u⁰_2i−1) = 5 − 6i, 1 ≤ i ≤ n + 1 2



f (u⁰2i) = 6n − 6i + 3, 1 ≤ i ≤ n − 1 2



f (u⁰⁰2i−1) = 3 − 6i, 1 ≤ i ≤ n + 1 2



f (u⁰⁰2i) = 6n − 6i + 1, 1 ≤ i ≤ n − 1 2



f (v2i−1) =







−3n − 6i + 4, 1 ≤ i ≤ⁿ⁺¹₂ and n is odd 3n − 6i + 5, 1 ≤ i ≤ⁿ₂ and n is even

f (v2i) =







3n − 6i + 2, 1 ≤ i ≤ⁿ⁻¹₂ and n is odd

−3n − 6i + 1, 1 ≤ i ≤ⁿ₂ and n is even

f (v⁰_2i−1) =







3n − 6i + 6, 1 ≤ i ≤ⁿ⁺¹₂ and n is odd

−3n − 6i + 5, 1 ≤ i ≤ⁿ₂ and n is even

f (v2i⁰ ) =







−3n − 6i + 2, 1 ≤ i ≤ⁿ⁻¹₂ and n is odd 3n − 6i + 3, 1 ≤ i ≤ⁿ₂ and n is even

f (v⁰⁰_2i−1) =







3n − 6i + 4, 1 ≤ i ≤ⁿ⁺¹₂ and n is odd

−3n − 6i + 3, 1 ≤ i ≤ⁿ₂ and n is even

f (v2i⁰⁰) =







−3n − 6i, 1 ≤ i ≤ⁿ⁻¹₂ and n is odd 3n − 6i + 1, 1 ≤ i ≤ⁿ₂ and n is even f (uiui+1) = 6n − 6i, 1 ≤ i ≤ n − 1

f (uiu⁰i) = 6n − 6i + 4, 1 ≤ i ≤ n f (uiu⁰⁰_i) = 6n − 6i + 2, 1 ≤ i ≤ n f (vivi+1) = −6i, 1 ≤ i ≤ n − 1

f (viv⁰i) = −6i + 4, 1 ≤ i ≤ n f (viv⁰⁰_i) = −6i + 2, 1 ≤ i ≤ n f

 un+1

2 vn+1 2



= 0, if n is odd and f

uⁿ

2+1vⁿ

2



= 0, if n is even.

It can be verified that f is a super pair sum labeling and hence Hn S2 is a super pair sum graph. For example, a super pair sum labeling of H5 S2 and H6 S2 are shown in Figure 4.

bbbbb bbbbb bbbbbbbbbb bbbbbbbbbb

-1 -3 27 25 -7

-9 21

19 -13

-15 28 26 22 20

16 14

10 8

4 2

29 24

-5 18

23 12

-11 6

17 0

15 13 -19 -21 9

7 -25 -27 3 1 -2 -4 -8

-10 -14

-16 -20

-22 -26

-28 -17

-6 11

-12 -23

-18 5

-24 -29 H

⊙ S

bbbbb bbbbb bbbbbbbbbb

bbbbbbbbbb

-1 -3 33 31 -7

-9 27

-13 25

-15 34 32 28 26

22 20

16 14 10 8

35 30

-5 24

29 18

-11 12

23 0

-19 -21 15 13 -25

-27 9 7 -31 -33 -2 -4 -8

-10 -14

-16 -20

-22 -26

-28 17

-6 -23

-12 11

-18 -29

-24 5

H

⊙ S

b b bb

bb

21 19

-17 4

2 -35

3

1 -32

-34

6 -30

Figure 4.

Theorem 2.4. The disconnected graph 2Hnfor n ≥ 3 is a super pair sum graph.

Proof. Let u⁰1, u⁰2, . . . , u⁰nand v⁰1, v⁰2, . . . , v⁰nbe the vertices of the first copy of Hnand u⁰⁰1, u⁰⁰2, . . . , u⁰⁰nand v1⁰⁰, v⁰⁰2, . . . , vn⁰⁰be the vertices of the second copy of Hn. The graph 2Hn has 4n vertices and 4n − 2 edges. Define f : V (2Hn) ∪ E(2Hn) → {±1, ±2, . . . , ±4n − 1} as follows:

f (u⁰i) =







i, 1 ≤ i ≤ n and i is odd

−(4n + 1) + i, 1 ≤ i ≤ n and i is even

f (v⁰i) =



















−(3n + 1) + i, if n is odd, 1 ≤ i ≤ n and i is odd n + i, if n is odd, 1 ≤ i ≤ n and i is even n + i, if n is even, 1 ≤ i ≤ n and i is odd

−(3n + 1) + i, if n is even, 1 ≤ i ≤ n and i is even

f (u⁰⁰i) =







2n + i, 1 ≤ i ≤ n and i is odd

−(2n + 1) + i, 1 ≤ i ≤ n and i is even

f (v⁰⁰i) =



















−(n + 1) + i, if n is odd, 1 ≤ i ≤ n and i is odd 3n + i, if n is odd, 1 ≤ i ≤ n and i is even 3n + i, if n is even, 1 ≤ i ≤ n and i is odd

−(n + 1) + i, if n is even, 1 ≤ i ≤ n and i is even

f (u⁰iu⁰i+1) = −4n + 2i, 1 ≤ i ≤ n − 1 f (v⁰_iv_i+1⁰ ) = −2n + 2i, 1 ≤ i ≤ n − 1 f (u⁰⁰iu⁰⁰i+1) = 2i, 1 ≤ i ≤ n − 1

f (v⁰⁰ivi+1⁰⁰ ) = 2n + 2i, 1 ≤ i ≤ n − 1 f

u⁰n+1 2

v⁰n+1 2



= −2n if n is odd f

u⁰⁰n+1 2

v⁰⁰n+1 2



= 2n if n is odd f

u⁰n 2+1v⁰n

2



= −2n if n is even f

u⁰⁰ⁿ

2+1v⁰⁰ⁿ

2



= 2n if n is even.

Thus, f is a super pair sum labeling and hence 2Hn is a super pair sum graph for n ≥ 3. For example, a super pair sum labeling of 2H7 and 2H8 are shown in Figure 5.

bbbbbbb bbbbbbb bbbbbbb bbbbbbb

b b

1 -26 -27

-24 3

-22 -25

-20 5

-18 -23

-16 7

-14 -21 -12

9 -10

-19 -8

11 -6

-17 -4

13 -2

-15

1 -30 -31

-28 3

-26 -29

-24 5

-22 -27

-20 7

-18 -25

-16 9 -14

-23 -12

11 -10

-21 -8

13 -6

-19 -4

15 -2

-17 2H8

bbbbbbb bbbbbbb

15 2 -13

4 17

6 -11

8 19

10 -9

12 21

14 -7 16

23 18

-5 20

25 22

-3 24

27 26

-1 2H7

bbbbbbb bbbbbbb

b b

17 2 -15

4 19

6 -13

8 21

10 -11

12 23

14 -9

16 25 18

-7 20

27 22

-5 24

29 26

-3 28

31 30

-1

Figure 5.

References

[1] J.A.Gallian, A dynamic survey of graph labeling, The Electronic J. Combin., (17)(2010).

[2] F.Harary, Graph Theory, Narosa publishing House, New Delhi, (1998).

[3] R.Ponraj, J.Vijaya Xavier Parthipan and R.Kala, Some results on pair sum labelings of graphs, International Journal of Mathematical Combinatorics, (4)(2010), 53-61.

[4] R.Ponraj, J.Vijaya Xavier Parthipan and R.Kala, A note on pair sum graphs, Journal of Scientific Research, (3)(2)(2011), 321-329.

[5] Sin-Min Lee, Elo leung and Ho Kuen Ng, On super vertex-graceful unicyclic graphs, Czechoslovak Math. J., (59)(134)(2009), 1-22.

[6] R.Vasuki and S.Arockiaraj, Super pair sum labeling of graphs, (communicated).