Duplication of some graph elements and absolute mean graceful labeling

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International Journal of Mathematics Trends and Technology (IJMTT) – Volume X Issue Y- April 2019

D

uplication of some graph elements and

a

bsolute mean graceful labeling

1_{H. P. Chudasama and} 2_{V. J. Kaneria} 1 _{Government Polytechnic, Rajkot}₋₃₆₀₀₀₃

2 _{Department of Mathematics, Saurashtra University, Rajkot}₋_{360005, India.} 1_{[email protected],} 2 _{[email protected]}

Abstract :

In the present paper, we derived absolute mean graceful labeling for some graphs obtained by duplication of graph elements in complete bipartite graphKm,n, cycle Cn, pathPn and swastik graphSwn.

Key Words :

Duplication of a vertex by vertex, duplication of a vertex by edge, duplication of edge by a vertex, α - absolute mean graceful graph, swastik graph.

AMS subject classification(2010) :

05C78.

1. Introduction :

All the graphs, going to be discussed in the present paper, are finite, simple and undirected. Let G be a (p, q) graph. For a graph G = (V, E) a function having domain V or E or V ∪ E is known as graph labeling for G. Throughout this paper, complete bipartite graph will be denoted as Km,n with m part M = {u1, u2, ..., um} and n part N = {v1, v2, ..., vn}.

i.e. V(Km,n) = M ∪N. Concept of graph labeling was invented by Rosa

[1]. Kaneria and Chudasama [2] gave more freedom to graceful labeling and introduced absolute mean graceful labeling. Kaneria and Chudasama [3] proved that graceful labeling is preserved even after duplication by vertex or edge in Km,n. In this present work, we obtained absolute mean graceful labeling for

various graphs obtained by duplication of same graph elements. We also proved that Swastik graph is absolute mean graceful which was earlier proved graceful graph by Kaneria and Makadia [4]. For conceptual study and notations, we referred Gallian [5] and Harary [6].

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Duplication of a vertex v of a graph G is the graph G0 by adding a new vertexv0 (duplicant of v) such that NG0(v0) =N_G(v) =N_G0(v). i.e. v0 is

adjacent with all vertices of Gwhich are adjacent to v inG.

Duplication of a vertex v by a new edge e = v0v00 in a graph G produces a new graph G0 = (V(G)∪ {v0, v00}, E(G)∪ {v0v00, v v0, v v00}). Duplication of an edge e =uv by a new vertex w in a graph G produces a new graphG0 = (V(G)∪ {w}, E(G)∪ {uw, vw}). i.e. NG0(w) = {u, v}.

Swastik graph is an union of four copies on C4n. If vi,j (∀ i = 1,

2, 3, 4; ∀ j = 1, 2,..., 4n) be vertices of ith copy of C₄(i_n), the we shall combine v1,4t and v2,1; v2,4t and v3,1; v3,4t and v4,1; v4,4t and v1,1 by four vertices. So graph seems like a plus sign. If we bend branches of graph toward clockwise at the middle, then the graph looks a swastik which is denoted asSwn of n size,

wheren ∈N − {1}. Clearly, |V(Swn)|= 16(n)−4 and |E(Swn)|= 16(n).

Definition : α - absolute mean graceful graph

A function f is called an absolute mean graceful labeling of a graph G = (V, E), if f : V(G) → {0,±1,±2, ...,±q} is injective and the induced

function f∗ : E(G) → {1,2, ..., q} defined as f∗(e) =

_|

f(u)−f(v)|

2

is

bijective for every edge e = (u, v) ∈ E(G). A labeling f is said to be α - labeling, if there exists an integer k such that for each edge uv either f(u) ≤ k <| f(v) | or f(v) ≤ k <| f(u) | , ∀u, v ∈ V(G). The graph which holds absolute mean graceful labeling and α - labeling is called α - absolute mean graceful graph.

2. Main Results :

Theorem 2.1 : Duplication of any vertex in Km,n is α - absolute mean

graceful graph.

Proof: Let G be a graph obtained by duplication of one vertex of Km,n. It

is obvious that G is either Km+1,n or Km,n+1. Since Km,n, ∀ m, n ∈ N is

absolute mean graceful graph proved by Kaneria and Chudasama [2],Km+1,n

and Km,n+1 both are absolute mean graceful graphs. Hence, G is absolute mean graceful graph. It also satisfies α-labeling. So, G is α - absolute mean

graceful graph.

Theorem 2.2 : Duplication of all the vertices ofm-part orn-part inKm,n is

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International Journal of Mathematics Trends and Technology (IJMTT) – Volume X Issue Y- April 2019 α - absolute mean graceful graph.

Proof: Let G be a graph obtained by duplication of all the vertices of M ={u1, u2, ..., um} in Km,n. ThenG=K2m,n.

Since Km,n, ∀ m, n∈ N is absolute mean graceful graph proved by

Kaneria and Chudasama [2], K2m,n is absolute mean graceful graph. Hence,

G is absolute mean graceful graph. It also satisfies α - labeling. So, G is α

-absolute mean graceful graph.

Theorem 2.3 : Duplication of any vertex of Km,n by an edge is absolute

mean graceful graph, but notα - absolute mean graceful.

Proof: Let G be a graph obtained by duplication of vertex ut, (1 ≤ t ≤ m)

of m-part of Km,n by an edge u0tu00t.

i.e. V(G) ={u1, u2, ..., um, v1, v2, ..., vn} ∪ {u0t, u

00

t} and

E(G) ={uivj /1≤i≤m,1≤j ≤n} ∪ {utu0t, utut00, u0tu00t}.

i.e. p=|V(G)|=m+n+ 2 andq =mn+ 3.

Definef :V(G)→ {0, ±1, ±2, ..., ±q}as follows:

f(ui) =

(

q, i= 1

f(ui−1)−2n, i= 2, 3, ..., m.

f(vj) =

(

−q, i= 1

f(vj−1) + 2, i= 2, 3, ..., n.

Case I : If f(ut)≥0, thenf(u 0

t) =f(ut)−5 and f(u 00

t) =f(ut)−3.

Case II : If f(ut)<0, thenf(u 0

t) =f(ut) + 5 and f(u 00

t) =f(ut) + 3.

Similarly, it can be easily proved for duplication of vertex vt (1 ≤

t ≤ n) of n-part of Km,n by an edge v 0 tv

00

t by replacing ut, u 0 t, u

00

t as vt, v 0 t, v

00 t

respectively. It is clear that edge labeling function f∗ : E(G) → {1,2, ..., q}

defined as f∗(e) =

|f(u)−f(v)|

2

is bijective, for every edge e = (u, v) ∈

E(G). Therefore,Gis absolute mean graceful graph, but it is not α- absolute

mean graceful graph.

Theorem 2.4 : Duplication of any edge of Km,n by a new vertex is α

-absolute mean graceful graph.

Proof: Let G be a graph obtained by duplication of an edge ukvl by a

vertex w in Km,n. So that V(G) = V(Km,n)∪ {w} and E(G) = E(Km,n)∪

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{ukw, wvl}.i.e. p=|V(G)|=m+n+ 1 and q =mn+ 2. Define f :V(G)→ {0, ±1, ±2, ..., ±q} as follows:

f(ui) =



 

 

q−2ni, i < k q, i=k q−2n(i−1), i > k.

f(vj) =



 

 

2(i+ 1)−q, j < l 2−q, j =l 2i−q, j > l.

f(w) = 1−q. It is clear that edge labeling function f∗ :E(G)→ {1,2, ..., q}

|f(u)−f(v)|

2

E(G). Therefore, G is absolute mean graceful graph. It also satisfies α -labeling. So,G is α - absolute mean graceful graph.

Theorem 2.5 : The graph obtained by duplication of both the verticesu1, u2 from 2-part in K2,n, ∀n ∈N − {2} by edges is absolute mean graceful, but it

is notα - absolute mean graceful graph.

Proof: LetG be a graph obtained by duplication of both the vertices u1, u2 from 2-part inK2,n, ∀n ∈N− {2} by edges e1 =u

0

1u

00

1 and e2 =u

0

2u

00

2. i.e. V(G) =V(K2,n)∪ {u

0

1, u

00

1, u

0

2, u

00

2}and E(G) =E(K2,n)∪{u1u

0

1, u

0

1u

00

1, u

00

1u1, u2u

0

2, u

0

2u

00

2, u

00

2u2}. i.e. p=|V(G)|= n+ 6 andq = 2n+ 6.

f(u0₁) = q−11, f(u00₁) = q−9, f(u0₂) = −2, f(u00₂) = 1, f(u1) = q, f(u2) =q−2n f(vi) = −q+ 2(i−1), ∀i= 1, 2, ..., n.

It is clear that edge labeling functionf∗ :E(G)→ {1,2, ..., q}defined

as f∗(e) =

_|

f(u)−f(v)|

2

is bijective, for every edge e = (u, v) ∈ E(G).

Therefore, G is absolute mean graceful graph. It does not satisfy α-labeling. So, G is absolute mean graceful graph, but not α - absolute mean graceful

graph.

Illustration 1: Duplication of both the vertices from 2-part ofK2,6 by edges is absolute mean graceful graph. It is clear thatp= 12 and q = 18.

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7 9 -2 1

-18 -16 -14 -12 -10 -8

18 6

Theorem 2.6 : Duplication of any vertex in cycle Cn, where n≡0 (mod2)

isα - absolute mean graceful graph.

Proof: Without loss of generality, let us assume that G is a graph obtained by duplication of vertex v1 of cycle Cn, where n ≡ 0 (mod2) by a new

vertex u such that N(v1) = N(u). Assign the other vertices v2, v3,..., vn

of G in anticlockwise direction such that V(G) = V(Cn)∪ {u} and E(G) =

E(Cn)∪ {uvn, uv2}. Observe that p=n+ 1 andq =n+ 2

f(vi) =



     

     

q, i= 1

(−1)i+1_[_|_f(v

i−1)| −2 ], i= 2, 3, ..., q 2+ 1 (−1)i+1_{5, i}₌ q

2 + 2 (−1)i+1_[_|_f(v

i−1)|+2 ], i= q 2 + 3,

q

2 + 4, ..., n.

f(u) = 4−q. It is clear that edge labeling function f∗ : E(G) → {1,2, ..., q}

|f(u)−f(v)|

2

E(G). Therefore,Gis absolute mean graceful graph. It also satisfiesα-labeling.

So, Gis α - absolute mean graceful graph.

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Illustration 2: Duplication of any vertex of cycle C8 by a new vertex is α - absolute mean graceful graph. It is clear that p= 9 and q= 10.

10

7

-5

2 -4

6 -9

-8 -6

Theorem 2.7 : Duplication of any vertex in path Pn,∀n ∈N isα - absolute

mean graceful graph.

Proof: Let G be a graph obtained by duplication of any vertex vk (1 ≤

k≤

n 2

) of path Pn by vertex u such thatV(G) = V(Pn)∪ {u} and

E(G) =

(

E(Pn)∪ {v2u or vn−1u}, if k = 1 or n E(Pn)∪ {vk−1u, vk+1u}, otherwise.

So thatp=n+ 1 and q=norn+ 1. Definef :V(G)→ {0, ±1, ±2, ..., ±q}

as follows:

f(vi) =



      

      

q, i= 1

(−1)i+1[|f(vi−1)| −1 ], i= 2, 3, ..., k+ 2

−f(vi−1), i=k+ 3 (−1)i+1[|f(vi−1)| −2 ], i=k+ 4

(−1)i+1[|f(vi−1)| −1 ], i=k+ 5, k+ 6, ..., n.

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f(u) =







−q, k= 1

(−1)k_[_|_f_(v

k+1)| −2 ], k= 2, 3, ...,

n 2

.

It is clear that edge labeling functionf∗ :E(G)→ {1,2, ..., q}defined

as f∗(e) =

_|

f(u)−f(v)|

2

is bijective, for every edge e = (u, v) ∈ E(G).

Therefore,G is absolute mean graceful graph. It also satisfies α-labeling. So,

Gis α - absolute mean graceful graph.

Theorem 2.8 : Every swastik graph Swn, ∀n ≥ 4 is α - absolute mean

graceful graph.

Proof: Let vi,j (∀i= 1, 2, 3, 4; ∀j = 1, 2, ..., 4n) be vertices of ith copy

of Ci

4n. By combining vertices v1,4t and v2,1; v2,4t and v3,1; v3,4t and v4,1; v4,4t

andv1,1as four vertices, we can get the Swastik graph by bending and shaping. It is clear thatp= 16(n)−4 and q= 16(n).

Define f :V(Swn)→ {0, ±1, ±2, ..., ±q} as follows:

f(vi,j) =

                                                                        

4, i= 1 and j = 1

−q, i= 1 and j= 2

(−1)j+1[|f(vi,j−1)| −1 ], i= 1 and j = 3, 4, ..., 4n−1

−3, i= 1 and j = 4n

−(q−4n+ 3), i= 2 and j = 2

(−1)j+1[|f(vi,j−1)| −1 ], i= 2 and j = 3, 4, ..., 4n−1 0, i= 2 and j = 4n

q

2+ 4, i= 3 and j = 2

−[f(vi,j−1) + 4 ], i= 3 and j = 3, 5

|f(vi,j−1)| −6, i= 3 and j = 4, 6

−[|f(vi,j−1) + 1 ], i= 3 and j= 7

(−1)j[|f(vi,j−1)| −2 ], i= 3 and j= 8, 2n+ 5, 2n+ 7 (−1)j[|f(vi,j−1)| −1 ], i= 3 and j= 9, 10, ..., 4n−1;

j 6= 2n+ 5, 2n+ 7 (−2), i= 3 and j= 4n

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f(vi,j) =



         

         

q

4 + 6, i= 4 and j= 2

(−1)j[|f(vi,j−1)| −1 ], i= 4 and j = 3, 4, ..., 4n−1; j 6= 6, 7, 8, 2n+ 3, 2n+ 4 [f(vi,j−1)−4 ], i= 4 and j = 6

−[|f(vi,j−1) + 1 ], i= 4 and j = 7 [|f(vi,j−1)−2 ], i= 4 and j= 8, 2n+ 3, 2n+ 4.

It is clear that induced edge labeling functionf∗ :E(G)→ {1,2, ..., q}

_|

f(u)−f(v)|

2

E(G). Therefore, G is absolute mean graceful graph. It also satisfies α -labeling. So,G is α - absolute mean graceful graph.

Illustration 3: α - absolute mean graceful labeling in Sw4. It is clear that p= 60 and q= 64.

-64 63

-62 61 -60 59 -58 57 -56 55 -54

53

-52

51

-3 -51 50 -49

48 -47

46 -45

44

-43

42

-41

40 -39 ₃₈ 0

36

-40

34 -38 32 -33

31 -30 29 _-28 ₂₇

-25 24 -22 -2

22 -21 20

-19

15

-16 14

-13 12 -10

8 -7

6 -5 4

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3. References :

[1] A. Rosa, On Certain Valuation of Graph Theory of Graphs (Rome July 1966), Goden and Breach, N. Y. and Paris, 1967, pp. 349-355.

[2] V. J. Kaneria and H. P. Chudasama, Absolute mean graceful labeling in various graphs, Int. J. of Mathematics and its applications, Vol. 5, No. 4-E (2017), pp. 723-726.

[3] V. J. Kaneria and H. P. Chudasama, Graceful Labeling in the Context of Duplication of Some Graph Elements in Km,n, Int. J. of Mathematics

and its applications, Vol. 5, 2017, pp. 53-55.

[4] V. J. Kaneria and H. M. Makadia, Graceful Labeling for Swastik Graph, Int. J. of Mathematics and its applications, Vol. 3, No. 3-D (2015), pp. 25-29.

[5] J. A. Gallian, A Dynamic Survey of Graph Labeling, The Electronics Journal of Combinatorics, 18(2015),] DS6.

[6] F. Harary, Graph Theory,Narosa Publishing House, New Delhi, 2001.