for graph operations

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J. Sci. Res. 3 (2), 291-301 (2011)

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On Study of Vertex Labeling of Graph Operations M. A. Rajan1*, V. Lokesha2, and K. M. Niranjan3 1

Innovation Labs, Tata Consultancy Services Limited, Bangalore, India

2

Department of Mathematics, Acharya Institute of Technology, Bangalore -560090, India 3

Department of Mathematics, S.J.M.I.T, Chitradurga-577 502, India

Received 10 August 2010, accepted in final revised form 23 March 2011 Abstract Vertices of the graphs are labeled from the set of natural numbers from 1 to the order of the given graph. Vertex adjacency label set (AVLS) is the set of ordered pair of vertices and its corresponding label of the graph. A notion of vertex adjacency label number (VALN) is introduced in this paper. For each VLS, VLN of graph is the sum of labels of all the adjacent pairs of the vertices of the graph. ℵ is the maximum number among all the VALNs of the different labeling of the graph and the corresponding VALS is defined as maximal vertex adjacency label set MVALSℵ . In this paper ℵ for different graph operations are discussed. Keywords: Subdivision; Graph labeling; Direct sum; Direct product. © 2011 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. doi:10.3329/jsr.v3i26222 J. Sci. Res. 3 (2), 291-301 (2011)

1. Introduction The vertex natural labeling of graphs is introduced in ref. [1-3]. Research in the graph enumeration and graph labeling started way back in 1857 by Arthur Cayley. Graph enumeration is defined as counting number of different graphs of particular type, subgraphs, etc. with graph variants (the number of vertices and edges of the graph). Labeling of graph is assigning labels to the vertices or edges of a graph. Most graph labeling concepts trace their origins to labeling presented by Alex Rosa [1]. Some of graph labeling methods are graceful labeling, harmonious labeling, and coloring of graphs. For the detailed survey on graph labeling see ref. [4]. Graph labeling and enumeration finds the application in chemical graph theory, social networking and computer networking. For example, Cayley demonstrated that the number of different trees of n vertices is analogues to number of isomers of the saturated hydrocarbon with n carbon items C n H 2n+2 . More such applications can be found in ref. [1]. This paper is an *

Correponding author: [email protected]

292

On Study of Vertex Labeling

extension of our previous work [5], in which vertex labeling for graph operations are studied. The following definitions are used abundantly in this paper. Vertex natural labeling of a graph: A vertex natural labeling of G is a mapping function l which assigns each vertex v of G, an unique number l(u) from the set of natural numbers N = {1,2,3,......, p}. That is all the vertices are having distinct labels from 1 to p. Thus l is bijective. So there are p! sets of different labeling of a given graph. Each such label set is called vertex adjacency label set (VALS). Vertex adjacency labeling number (VALN) for each VALS is defined as the sum of labels of all the adjacent pairs of the vertices of the graph, which is given by

∑ l(u) + l(v) . Let the set {VALS1,VALS2,.........,VALS p!}be

for each edge(u,v)∈E(G )

all the VALSs of the given graph G( p, q) and the set {VALN1 , VALN 2 ,........., VALN p!} be

set of corresponding VALNs. Then ℵ(G ) = Max(VALN1 , VALN 2 ,........., VALN p! ) is the maximum vertex label number of a given graph G( p, q) and the corresponding VALS is called maximal vertex adjacency label set(MAVLS). Similarly vertex non adjacency labeling number (VNALN) for each VALS is defined as sum of labels of all non adjacent pairs of the vertices of the graph, which is given by

∑ l(u) + l(v) .

for each edge(u,v)∉E(G )

Subdivision of a graph: The subdivision of graph of a graph G denoted by S(G) is a graph obtained from G by replacing each of its edge by two series edges by introducing a new vertex into each edge of the G. See the ref. [7] for more details. Direct Sum of two graphs: Direct sum of two graphs G1(V1, E1) and G2(V2 , E2 ) is denoted

by G1 ⊕ G2 where V1 ∩ V2 = φ is the graph G(V, E ) for which V = V1 + V2 and E = E1 + E2 .

Direct product of two graphs: The direct product of two graphs G1(V1, E1) and G2(V2 , E2 ) is denoted by G1 ⊗ G2 is the graph obtained from G1 ⊕ G2 by joining every vertex of G1(V1, E1) with every vertex of G2(V2 , E2 ) . Total number of links in complete product of two graphs is E1 + E2 + V1 V2 . This work is an extension of our paper [5] in which ℵ is computed for various graph operation which includes subdivision, direct sum and product operations. The paper is organized into several sections. In section 2, important results and observations which are useful for deriving the new results are described. In section 3, new problems are stated and results are provided for these problems. Conclusions are given in section 4. 2. Known Results Most of the graph theory definitions are found in literature [6-8]. We recall some essential results which are required in a sequel from ref. [5].

M. A. Rajan et al. J. Sci. Res. 3 (2), 291-301 (2011)

293

i.

For complete graph, K p , any order vertex natural label of graphs gives ℵ . That

ii.

is all the p! Vertex labeling of graphs yields same ℵ value. For a G( p, r ) is r-regular graph, any order vertex natural label of graphs gives

iii.

ℵ. For any graph G, by assigning the labels from the label set {p, p − 1,....,1} to the

{v , v ,....v }

VALS,

iv. v.

1

p

respectively, where

d1 ≥ d2 ≥ .... >= d p one of the

MVALS and corresponding MVALN is ℵ(G) . For any complete graph with K p , ℵ K p = ( p − 1)p( p + 1) . 2 is a complete bipartite graph with If G = K m,n

( )

ℵ( K= m,n ) vi.

2

m≥n,

then

1 [ m(m + n)(m + n + 1) - (m - n)m(m + 1)]. 2

If G( p, r ) is a r-regular graph , then

p( p + 1) . 2 p( p + 1) . (b). ℵ G ( p, r ) = ( p − (1 + r )) 2 (c). ℵ(G ( p, r )) + ℵ G ( p, r ) = ℵ(K p ) . (a). ℵ(G ( p, r )) = r

(

)

(

vii.

Let

)

G ( p, q ) is a graph, whose vertices have degree either m or n. Let p m

vertices have degree m and

pn vertices have degree n. Then

  p + 1  m   2    p + 1  p + 1  − (m − n ) n  ℵ(G ( p, q )) = m 2   2      p + 1  p + 1 n  − (n − m ) m     2   2 

for m = n for m  n for m  n

.

3. Results Theorem 1: Let H = G − v is a graph obtained from G, by removing any vertex v from v1, v2,....v p be the vertices of G such that d1 ≥ d2 ≥ .... >= d p and the new

G. Let

{

}

vertex sequence

v1 , v2 ,.. , v j −1 , vi j +1 , … , v p is unaltered. Let v = v j be the vertex,

which has degree

d j to be removed from G, then

294

On Study of Vertex Labeling

(a).

 j −1  ℵ(H ) ≤ ℵ(G) − ∑ dk + jd j  .  k =1 

 j −1

(b). ℵ(H ) = ℵ(G) − 

∑d

 k =1

k

 + jd j  − ∑ p − k + 1 − (v ,v )∈∑E(G),pk j k j

Proof: (a). From the data, the subgraph H is obtained from G by removing the vertex Let

the

vertices

of

G

v1 , v2 ,.. ,

v j −1 , v j

,

…

,

vi j .

v p are labeled as

p, p − 1,. p − 2,... p − j, p − j − 1...,1 respectively. So in H, the vertices, v1 , v2 ,.. , v j −1 , vi j +1 ,…,

v p are re-labeled as p, p − 1, p − 2.,... p − j − 1,...,1 respectively. Thus j −1 d has to ∑ i

be reduced from ℵ(G ) . Clearly the contribution of vertex zero. So

k =1

k

v j to the summation ℵ(H ) is

jd j needs to be reduced from ℵ(G) on removing the vertex v j from G, results in

removal of edges associated from it. This implies the contribution of labels to the summation from the adjacent vertices of v j needs to be removed. Thus

 j −1  ℵ(H ) ≤ ℵ(G) − ∑ dk + jd j  . k =1   Proof: (b). From the above proof, ℵ(H ) ≤ ℵ(G) −  ∑ dk + jd j  . To reduce it to equality, j −1

k =1



the RHS of the inequality further needs to be subtracted. The vertices which are adjacent to v j can be of two types: vertices with their labels higher or lower than that of v j . In the above inequality, for the adjacent vertices with their labels lower than that of

v j only

their degree are reduced by 1 and where as for the adjacent vertices with their labels. Theorem 2: Let S (K p ) is a graph obtained by subdivision operation on K p , then

 p    p + 1     + 1   + 1  + ( p − 3)  2   .      2    2

( ( ))

(a). ℵ S K = ( p − 1)  2  p  

Proof:

The

vertices

of

S (K p )

is

partitioned

{

}

into, V1 = v1, v2,...v p ,

  V2 = v p +1 , v p + 2 ,...v  p   with vertices of sets V 1 and V 2 have degree p − 1 and 2 p +  2  

M. A. Rajan et al. J. Sci. Res. 3 (2), 291-301 (2011)

295

p p + 1 respectively. ⇒ m = p − 1, n = 2, pm = p, pn = p p − 1 =  , p =   . Substituting    2 2   2  these values in result (VII),  p    p + 1     + 1    + 1 ℵ S K p = ( p − 1)  2   + ( p − 3)  2   .      2    2

( ( ))

  p + 1  (b). ℵ S K =   p  + p − 2   2  + 1 + ( p − 2 ) p + 1 . p   2     2          2

( ( ))

Proof:

The

vertices

of

S (K p )

is

partitioned

{

}

into, V1 = v1 , v2 ,...v p ,

  V2 = v p +1 , v p + 2 ,...v  p   with vertices of sets V 1 and V 2 have degree p − 1 and 2 p +  2   p p p p + 1 . Substituting respectively. ⇒ m =   + p − 2, n =  , pm =  , pn = p, p =  2  2  2   2          these values in the result (VII),   p + 1   + 1    p  p + 1  ℵ S (K p ) =    + p − 2   2   + ( p − 2 ) 2  2        2

(

)

Theorem 3: Let S (G ( p, r )) is a graph obtained by subdivision operation on r-regular graph G, then  p     p + 1     + 1 .  + 1 (a). ℵ(G( p, r )) = r    2   + (r − 2)  2         2   2

  p + 1  (b). ℵ S (G( p, r )) =  pr + p − 3  2  + 1 + ( p + r − 5) p + 1 .    2    2       2  

(

)

Proof: (a). The vertices of S (G ( p, r )) is partitioned into V1 = {v1 , v2 ,...v p },

296

On Study of Vertex Labeling

V2 = vp +1, vp + 2,...v pr  with vertices of sets V 1 and V 2 have degree r and 2 p+ 2   p + 1 . By substituting these respectively. ⇒ m = r, n = 2, pm = p, pn = pr , p =   2  2   values in result  p     p + 1  (VII), ℵ(S (G( p, r ))) = r  2  + 1 + (r − 2)  2  + 1 .             2   2 Proof:

(b).

The

vertices

of

S (G( p, r ))

is

partitioned

into, V1 = {v1 , v2 ,...v p },

V2 = vp +1, vp + 2,...v pr  with vertices of sets V 1 and V 2 have degree p+ 2 

and p(r + 2 ) − 3 , respectively. 2

⇒m=

( p − 2)(r + 2) + 1 2

( p − 2)(r + 2) + 1, p = pr , p = p, p =  p + 1 . p(r + 2) − 3, n = m  2  2 n 2 2  

By substituting these values in the result (VII),   p + 1    + 1  p + 1 .  ℵ S (G( p, r )) =  pr + p − 3  2   + ( p + r − 5)   2  2     2

(

)

Theorem 4: Let then

S (G( p, q)) is a graph obtained by subdivision operation on graph G,

(a). ℵ(S (G( p, q))) = ℵ(G( p, q)) + q(3q + 1)

(

)

(b). ℵ S (G ( p, q )) = ( p + q − 1) p + 1 + ( p + q − 3) pq +  q + 1  − ℵ(G ( p, q )) .  2   2        or   p + 1  q + 1   + ( p + q − 3) pq +    − q(3q + 1) . ℵ S (G( p, q)) + ℵ(S (G( p, q))) = ( p + q − 1) 2 2      Proof: Let v1, v2,...vp be the vertices of G such that d1 ≥ d 2 ≥ .... >= d p . Vertices of

(

)

{

}

{

S (G( p, q)) is partitioned into V1 = v1 , v2 ,...v p degrees of

{d ,...., d } i1

ip

and

V2 = {u1, u2,....uq}

}

and

V2 = {u1, u2,....uq} with their

{2,2,2,….} upto q, respectively. Label the vertices as

{1,2,3,...., q}

and

V1 = {v1 , v2 ,...v p }

as

M. A. Rajan et al. J. Sci. Res. 3 (2), 291-301 (2011)

{q + p, q + p − 1,......, q + 1} .

Then, the contribution of labels of the vertices of

V2 = {u1, u2,....uq} and V1 = {v1 , v2 ,...v p } to ℵ(S (G( p, q))) is p

∑ (q + i)d i =1

i p +1−i

297

q

∑ i * 2 = q(q + 1)

and

i =1

= 2q2 + ℵ(G( p, q)) respectively. By adding these two, we obtain,

ℵ(S (G( p, q))) = 2q2 + ℵ(G( p, q)) + q2 + q . Therefore, ℵ(S (G ( p, q ))) = ℵ(G ( p, q )) + q (3q + 1) .

(

)((

)

)

Note: In general, ℵ(Sn(G( p, q))) = ℵ(G( p, q)) + 2n − 1 q 3 2n − 1 q + 1

{v , v ,...v } be the vertices of G such that d ≥ d ≥ .... >= d . Vertices of S (G( p, q)) is partitioned into V = {v , v ,...v } and V = {u , u ,....u } with Proof: (b). Let

1

2

p

1

{p + q − 1 − d ,...., p + q − 1 − d } 1

their

degrees

{p + q − 3, p + q − 3,....... p + q − 3} the every vertex of of V1

= {v1 , v2 ,...v p }.

1

p

2

2

i1

1

p

2

q

ip

and

up to q terms respectively. Note that degrees of

V2 = {u1, u2,....uq} are greater than that of degrees of vertices Label

the

{p + 1, p + 2,......, p + q + 1}and

vertices

{

V1 = vi1, vi2 ,....vi p

}

V2 = {u1, u2,....uq}

of as

{p, p − 1,...1} .

as Then,

V2 = {u1, u2,....uq} and V1 = {v1 , v2 ,...v p } to

contribution of labels of the vertices of

(

2

)

ℵ S (G( p, q)) is q



i =1



 q + 1    and 2  

∑ ( p + i) * ( p + q − 3) = ( p + q − 3) pq +  ∑ i * (p + q − 1 − d p

i p +1−i

i =1

) = ( p + q − 1) p2+ 1 − S(G( p, q)) , respectively. 



By adding these two,

  p + 1  q + 1   + ( p + q − 3) pq +    − ℵ(G( p, q)) or ℵ S (G( p, q)) = ( p + q − 1)  2   2  

(

(

)

)

(

)

ℵ S ( G ( p, q ) ) +ℵ S ( G ( p, q ) ) =

Theorem 5: Let

  p + 1  q + 1   + ( p + q − 3)  pq +    − q ( 3q + 1) .  2   2  

( p + q − 1) 

G1( p, q) is a graph with p points and q edges.

298

On Study of Vertex Labeling (a). Let G = G1( p, q) ⊕ G1( p, q) , then ℵ(G ) = 2(2ℵ(G1 ) − q ) .

(b). Let G = G1( p, q) ⊗ G1( p, q) , then ℵ(G) = 4ℵ(G1) + 2 p3 + p2 − 2q .

Proof: (a). Graph G has 2p points and 2q edges. WLOG, the vertices

{v , v ,....v } of 1

2

p

the graph G 1 are labeled with {1,2,...., p} respectively and this set is MVALSℵ . Then the vertices of G

{v , v ,....v , v 1

2

p

v

p +1, p + 2

respectively and the vertices

,....v2 p} are labeled with {1,3,5,...2 p − 1,2,4,....2 p},

vi and vi + p are having same degree di . That is the vertices

of G 1 which is to the left and right of the operator ⊕ are assigned the odd and even labels respectively. This label set of G is MVALSℵ for G. Thus, ℵ(G ) =

p

∑ (2i − 1)d i =1

i

+

p

2p

∑ (2i )d

i = p +1

i

p

∑ ∑

=2

id i −

i =1

p

∑ id

di + 2

i =1

i

i =1

p

= 4∑ idi − 2q i =1

( )

Therefore, ℵ G

= 4ℵ(G1 ) − 2q = 2(2ℵ(G1 ) − q ) .

Proof: (b). Using the (a), and the degree of the vertex v i of G is 2p

p

ℵ(G ) = ∑ (2i − 1)(d i + p ) + i =1

∑ (2i )(d

i = p +1

p

= ∑ (2i − 1)(d i + p ) + i =1

+ p)

i

2p

∑ (2i )(d

i = p +1

di + p ,

i

+ p)

p

p

p

p

p

p

i =1

i =1

i =1

i =1

i =1

i =1

= ∑ 2id i + ∑ 2ip −∑ d i − ∑ p + ∑ (2i )d i + ∑ (2i ) p 3 2 Therefore, ℵ(G ) = 4ℵ(G1 ) + 2 p + p − 2q .

Theorem 6: If G1( p1, r1) and G2 ( p2, r2 ) are

r1 and r2 regular graphs respectively with

r1 ≥ r2 . (a). Let G G1 ( p1 , r1 ) + G2 ( p2 , r2 ) , then =

ℵ ( G ) = ℵ ( G1 ) +ℵ ( G2 ) + p1 p2 r1 or r1

p1 ( p1 + 1) p ( p + 1) + r2 2 2 + p1 p2 r1 . 2 2

M. A. Rajan et al. J. Sci. Res. 3 (2), 291-301 (2011) Proof: (a). The vertices and degree sequences of G1( p1, r1) and G2( p2, r2 ) be

{u , u ,....u } and {d , d ,..., d } , {d , d 1

2

p2

i1

i2

ip

1

j1

j2

,..., d j p

2

the

labels

{v , v ,....v } ,

} respectively and they are labeled with

{1,2,3,..., p1} and {1,2,3,..., p2 } , respectively. Note that[1], any label set is MVALSℵ . Since

299

1

2

p1

of a regular graph

r1 ≥ r2 , In G, the labels of the vertices of G 2 is unaltered, where as

of

vertices

{ p2 + 1, p2 + 2, p2 + 3,..., p2 + p1} ,

of

G1

{v , v ,....v }are 1

2

p1

changed

to

respectively. Then

p1

ℵ(G ) = ∑ r1 ( p2 + k ) + ℵ(G2 ) k =1 p1

p1

k =1

k =1

= ∑ r1k + ∑ r1 p2 + ℵ(G2 ) ℵ ( G ) = ℵ ( G1 ) +ℵ ( G2 ) + p1 p2 r1 or

p1 ( p1 + 1) p ( p + 1) . ,ℵ(G2 ) = r2 2 2 2 2 p ( p + 1) p ( p + 1) = ℵ ( G ) r1 1 1 + r2 2 2 + p1 p2 r1 . 2 2 By substituting, ℵ(G1 ) = r1

(b). Let G G1 ( p1 , r1 ) ⊗ G 2 (p2 , r2 ) , then =

ℵ ( G ) = (r1 + r2 )( p1 + p2 )( p1 + p2 + 1) / 2 , ℵ ( G ) = ( p1 + p2 − (1 + r1 + r2 ))( p1 + p2 )( p1 + p2 + 1) / 2 . Proof: (b). From the data, G is a regular graph with degree (iii), ℵ ( G ) = (r1 + r2 )( p1 + p2 )( p1 + p2 + 1) / 2 and

r1 + r2 . By using the result

ℵ ( G ) = ( p1 + p2 − (1 + r1 + r2 ))( p1 + p2 )( p1 + p2 + 1) / 2 . Theorem 7. If (a).

G1( p1, q1) is a graph and G2( p2, r2 ) is r2 -regular graph, then

= if G G1 ( p1 , q1 ) ⊕ G2 ( p2 , r2 )  ℵ ( G1 ) +ℵ ( G2 ) + p1 p2 r ,   ℵ( G ) =   p1 p2 3 2 2 ( , ) ( , ) ℵ G +ℵ G + p + p + r + = if G G p q ⊗ G p r ( 1) ( 2 ) ( 1 ) 2 1 1 1 2 2 2    2 

300

On Study of Vertex Labeling with r2 ≥ δ ( G1 ) .

(b).

ℵ ( G1 ) +ℵ ( G2 ) + 2 p2 q1 , if= G G1 ( p1 , q1 ) ⊕ G2 ( p2 , r2 )    ℵ( G ) = p + p + 2    1  2 ℵ G +ℵ G + p p = if G G ( p , q ) ⊗ G ( p , r ) ( ) ( ) 1 2 1 2 1 1 1 2 2 2    2    

with r2 ≤ δ ( G1 ) . Proof: (a). case (i): r2 ≥ δ ( G1 ) . Let the vertices

{v , v ,....v }of G 1

2

p1

1

and vertices of G 2

{u , u ,....u } are labeled with 1

2

p2

{1,2,3,..., p1} and {1,2,3,..., p2 } respectively, which gives ℵ(G1 ) and ℵ(G2 ) for G 1 and G 2 respectively. To obtain ℵ(G ) , In G, the vertices of the G1( p1, q1) and G2( p2, r2 ) are labeled with {1,2,3,..., p1} and

{ p1 + 1, p1 + 2, p1 + 3,..., p1 + p2 } , respectively.

Let G = G1 ⊕ G2 . Thus to ℵ(G ) , the contribution of G1 and G2 is ℵ(G1 ) and ℵ ( G2 ) + ∑ d ( ui ) p1 p2

i =1

respectively.

∴ ℵ ( G ) = ℵ ( G1 ) +ℵ ( G2 ) + p1 ∑ d ( ui ) p2

i =1

∴ ℵ ( G ) = ℵ ( G1 ) +ℵ ( G2 ) + p1 p2 r . Proof: (b). If G = G1 ⊗ G2 . Then degrees of the vertices of G 1 and G 2 are increased by p 2 and p 1 , respectively. Thus in ℵ(G ) , the contribution of the vertex v, u of G 1 and G 2 is l ( v ) ( d ( v ) + p2 ) , ( l ( u ) + p1 ) ( r + p1 ) , respectively.

∴ ℵ ( G ) = ℵ ( G1 ) + ∑ l ( vi ) p2 +ℵ ( G2 ) + ∑ l ( ui ) p1 + p1 ∑ ( r + p2 ) . p1

=i 1

p2

p2

=i 1 =i 1

 p ( p + 1)   p2 ( p2 + 1) ∴ ℵ ( G ) = ℵ ( G1 ) +ℵ ( G2 ) + p2  1 1  + p1  2 2    pp ∴ ℵ ( G ) = ℵ ( G1 ) +ℵ ( G2 ) + 1 2 ( p1 + 3 p2 + 2r + 2 ) . 2 Case ii: r2 ≤ δ ( G1 ) . Let the vertices v1, v2,....v p1 of G1 and vertices

{

}

 .  + p1 p2 ( r + p2 ) 

{u , u ,....u }of 1

2

p2

G 2 are labeled with {1,2,3,..., p1} and {1,2,3,..., p2 } respectively, which gives ℵ(G1 ) and ℵ(G2 ) for G 1 and G 2 , respectively. To obtain ℵ(G ) , in G, the vertices of the G2( p2, r2 )

M. A. Rajan et al. J. Sci. Res. 3 (2), 291-301 (2011) and G1( p1, q1) are labeled with {1,2,3,..., p2 } and respectively.

301

{p2 + 1, p2 + 2, p2 + 3,..., p2 + p1} ,

Let G = G1 ⊕ G2 . p1

Thus in ℵ(G ) , the contribution of G 1 and G 2 is ℵ(G1) + ∑ p2d (vi ) , ℵ(G2 ) and respectively. i =1

∴ ℵ(G ) = ℵ(G1) + ℵ(G 2) + 2 p2 q1 .

b) Let G = G1 ⊗ G2 . Then degrees of the vertices of G1 and G 2 are increased by

p2 and

p1 respectively. Thus in ℵ(G ) , the contribution of the vertex v, u of G 1 and G 2 is l (v )(d (v ) + p2 ), l (u )(r + p1 ) , respectively. ∴ ℵ ( G ) = ℵ ( G1 ) + ∑ l ( vi ) p2 +ℵ ( G2 ) + ∑ rp2 p1

p1

=i 1 =i 1

 p ( p + 1)   p2 ( p2 + 1)  ∴ ℵ ( G ) = ℵ ( G1 ) +ℵ ( G2 ) + p2  1 1  + p1  2 2      p + p2 + 2  . ∴ ℵ ( G ) = ℵ ( G1 ) +ℵ ( G2 ) + p1 p2  1  2   4. Conclusion In this paper, ℵ for the various graph operations are derived. The graph labeling finds applications in resource allocation, channel allocation in computer networks and communication. Exploring the application of graph labeling on graph operation is under way. As part of continuity to this work, a new label adjacency matrix representation of the graph and study of its spectra properties are under progress. References 1. A. Rosa, in: Theory of Graphs (Internat. Symposium, Rome, 1966) (Gordon and Breach, New York and. Dunod Paris, 1967) pp 349-355. 2. L. W. Beineke and S. M. Hegde, Discuss.Math. Graph Theory 21, 63 (2001). 3. A. F. Beardon, Austral. J. Combin. 30, 113 (2004). 4. J. A. Gallian, The electronic journal of combinatorics 17 (2010), #DS6. http://www.combinatorics.org/Surveys/ds6.pdf. 5. M. A. Rajan, V. Lokesha, and P.S. Ranjini, Int. e-Journal Math. Eng. 2 (2) 992 (2011) http://internationalejournals.com/vol2_iss2-a109.html 6. F. Harary, Graph Theory (Addison Wesley, Reading, Mass, 1969). 7. M. A. Rajan, V. Lokesha, and P. S. Ranjini, Proc. 23rd joint Iran-South Korea Jang Math. Soc., Iran, 23 (2010). 8. P. S. Ranjini, V. Lokesha, and M. A. Rajan, Int. J. Math. Sc. Eng. Appl. 4, 221 (2010).