ON THE HARMONIC INDEX OF GRAPH

Transactions on Combinatorics ISSN (print): 2251-8657, ISSN (on-line): 2251-8665 Vol. x No. x (2015), pp. xx-xx. c 2015 University of Isfahan ⃝ www.combinatorics.ir

www.ui.ac.ir

ON THE HARMONIC INDEX OF GRAPH OPERATIONS B. SHWETHA SHETTY, V. LOKESHA AND P. S. RANJINI

Communicated by Ivan Gutman

Abstract. The harmonic index of a connected graph G, denoted by H(G), is defined as ∑ 2 H(G) = uv∈E(G) du +d where dv is the degree of a vertex v in G. In this paper, expresv sions for the Harary indices of the join, corona product, Cartesian product, composition and symmetric difference of graphs are derived.

1. Introduction and Preliminaries

Throughout this paper we consider only simple connected graphs, i.e. connected graphs without loops and multiple edges. For a graph G, V (G) and E(G) denote the set of all vertices and edges, respectively. For a graph G, the degree of a vertex v is the number of edges incident to v and denoted by dv . The Harmonic index H(G) is vertex-degree-based topological index. This index first MSC(2010): Primary: 05C20; Secondary: 05C05. Keywords: Harmonic index, Graph operations. Received: 29 October 2014, Accepted: 1 December 2014. ∗Corresponding author.

1

2

Trans. Comb. x no. x (2015) xx-xx

B. S. Shwetha, V. Lokesha and P. S. Ranjini

appeared in [5], and was defined as[14], [16] ∑ H(G) = uv∈E(G)

2 du + dv

The connectivity index introduced in 1975 by Milan Randic [12], who has shown this index to reflect molecular branching. Randic index was defined as follows; ∑ 1 √ χ(G) = du dv uv∈E(G)

Ernesto Estrada et al [4], introduced atom-bond connectivity (ABC) index, which it has been applied up until now to study the stability of alkanes and the strain energy of cycloalkanes. This index is defined as follows: √ ∑ du + dv − 2 ABC(G) = du dv e=uv∈E(G)

For a vertex v in a connected nontrivial graph G, The eccentricity eccG (v) of a vertex v is the greatest geodesic distance between v and any other vertex. The Diameter d(G) of G is defined as d(G) = max{eccG (v)|v ∈ V (G)}. The Composition (also called lexicographic product [7]) G = G1 [G2 ] of graphs G1 and G2 with disjoint vertex sets V (G1 ) and V (G2 ) and edge sets E(G1 ) and E(G2 ) is the graph with vertex set V (G1 ) × V (G2 ) and (ui , vj ) is adjacent with (uk , vl ) whenever ui is adjacent with uk , or ui = uk and vj is adjacent with vl . The Cartesian product G1 × G2 of graphs G1 and G2 has the vertex set V (G1 × G2 ) = V (G1 ) × V (G2 ) and (ui , vj )(uk , vl ) is an edge of G1 × G2 if ui = uk and vj vl ∈ E(G2 ), or ui uk ∈ E(G1 ) and vj = vl [2]. For given graphs G1 and G2 we define their Corona product G1 oG2 as the graph obtained by taking |V (G1 )| copies of G2 and joining each vertex of the i-th copy with vertex vi ∈ V (G1 ). Obviously, |V (G1 oG2 )| = |V (G1 )|(1 + |V (G2 )|) and |E(G1 oG2 )| = |E(G1 )| + |V (G1 )|(|V (G2 )| + |E(G2 )|)[6]. A sum G1 + G2 of two graphs G1 and G2 with disjoint vertex sets V (G1 ) and V (G2 ) is the graph on the vertex set V (G1 ) ∪ V (G2 ) and the edge set E(G1 ) ∪ E(G2 ) ∪ {uv | u ∈ V (G1 ), v ∈ V (G2 )}. Hence, the sum of two graphs is obtained by connecting each vertex of one graph to each vertex of the other graph, while keeping all edges of both graphs[11]. The symmetric difference G1 ⊕ G2 of two graphs G1 and G2 is the graph with vertex set V (G1 ) × V (G2 ) and E(G1 ⊕ G2 ) = {(u1 , u2 )(v1 , v2 )|u1 v1 ∈ E(G1 ) or u2 v2 ∈ E(G2 ) but not both}. Obviously,|E(G1 ⊕ G2 )| = |E(G1 )||V (G2 )|2 + |E(G2 )||V (G1 )|2 − 4|E(G1 )||E(G2 )|

Trans. Comb. x no. x (2015) xx-xx

B. S. Shwetha, V. Lokesha and P. S. Ranjini

3

dG1 ⊕G2 (u, v) = |V (G2 )|dG1 (u) + |V (G1 )|dG2 (v) − 2dG1 (u)dG2 (v) [8]. In this paper, expressions for the Harmonic indices of the join, corona product, Cartesian product, composition and symmetric difference of graphs are derived[1],[3],[10],[13], [15],[9].

2. Harmonic index of graph operations

Theorem 2.1. Let G1 and G2 be two connected graphs with order n1 , n2 and size m1 , m2 respectively. Then {n 1 1 H(G1 [G2 ]) ≤ ABC(G2 ) + n1 R(G2 ) + n1 n2 m2 2 (1 + n2 ) 2 } (n2 )3 + ABC(G1 ) + (n2 )3 R(G1 ) + (n2 )2 m1 . 2 Proof. Let V (G1 ) = {u1 , u2 , . . . , un1 } and V (G2 ) = {v1 , v2 , . . . , vn2 } be a set of vertex for G1 and G2 respectively. By the definition of the composition of two graph one can see that, |E(G1 [G2 ])| = |E(G1 )||V (G2 )|2 + |E(G2 )||V (G1 )| dG1 [G2 ] (u, v) = |V (G2 )|dG1 (u) + dG2 (v) ∑

H(G1 [G2 ]) =

(ui ,vj ),(uk ,vl )∈E(G1 [G2 ]),(ui ,vj )̸=(uk ,vl )

∑

=

(ui ,vj ),(ui ,vl )∈E(G1 [G2 ]),j̸=l

2 dG1 [G2 ] (ui , vj ) + dG1 [G2 ] (ui , vl )

∑

+

(ui ,vj ),(uk ,vl )∈E(G1 [G2 ]),i̸=k

=

∑

∑

ui ∈V (G1 ) vj ,vl ∈E(G2 )

+

∑

∑

2 dG1 [G2 ] (ui , vj ) + dG1 [G2 ] (uk , vl )

2 |V (G2 )|dG1 (ui ) + dG2 (vj ) + |V (G2 )|dG1 (ui ) + dG2 (vl ) ∑

ui ,uk ∈E(G1 ) vj ∈V (G2 ) vl ∈V (G2 )

=

∑

∑

ui ∈V (G1 ) vj ,vl ∈E(G2 )

2 dG1 [G2 ] (ui , vj ) + dG1 [G2 ] (uk , vl )

2 |V (G2 )|dG1 (ui ) + dG2 (vj ) + |V (G2 )|dG1 (uk ) + dG2 (vl )

2 dG2 (vj ) + dG2 (vl ) + 2n2 dG1 (ui )

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Trans. Comb. x no. x (2015) xx-xx

∑

+

B. S. Shwetha, V. Lokesha and P. S. Ranjini

∑

∑

ui ,uk ∈E(G1 ) vj ∈V (G2 ) vl ∈V (G2 )

2 n2 dG1 (ui ) + n2 dG1 (uk ) + dG2 (vj ) + dG2 (vl )

= A1 + A2

(1)

where A1 , A2 are the sums of the above terms, in order. For any vertex u, v, w of a graph G and for any positive integer n1 , n2 , n3 1 n3 1 n1 n2 + ) ≤ ( + (n1 + n2 + n3 ) du dv dw + n2 dv + n3 dw )

1 (n1 +n2 +n3 ) (n1 du

with equality if and only if du = dv = dw . Therefore ∑ ∑ 2 A1 = dG2 (vj ) + dG2 (vl ) + 2n2 dG1 (ui ) ui ∈V (G1 ) vj ,vl ∈E(G2 )

≤

∑

2 2 + 2n2 {

= +

ui ∈V (G1 ) vj ,vl ∈E(G2 )

1 1 1 + n2 2 + 2n2 1 1 + n2

n2 + 1 + n2

∑ ∑

1 1 + n2

n2 + 1 + n2 =

∑

(

∑

1 } dG1 (ui )

∑

∑

ui ∈V (G1 ) vj ,vl ∈E(G2 )

∑

) 1 2 1 + − dG2 (vj ) dG2 (vl ) dG2 (vj )dG2 (vl )

1 dG2 (vj )dG2 (vl )

√

dG2 (vl ) + dG2 (vj ) − 2 dG2 (vl )dG2 (vj )

1 dG2 (vj )dG2 (vl ) }

√

ui ∈V (G1 ) vj ,vl ∈E(G2 )

∑

∑

ui ∈V (G1 ) vj ,vl ∈E(G2 )

ui ∈V (G1 ) vj ,vl ∈E(G2 )

∑

1 ( 1 1 2n2 ) + + 2 + 2n2 dG2 (vj ) dG2 (vl ) dG1 (ui )

∑

ui ∈V (G1 ) vj ,vl ∈E(G2 )

1 { 1 ≤ 1 + n2 2 + 2n2 +

∑

∑

1

ui ∈V (G1 ) vj ,vl ∈E(G2 )

1 { n1 n1 n1 n2 m2 } ABC(G2 ) + R(G2 ) + 1 + n2 2 + 2n2 1 + n2 1 + n2

Similarly A2 =

∑

∑

∑

ui ,uk ∈E(G1 ) vj ∈V (G2 ) vl ∈V (G2 )

2 n2 dG1 (ui ) + n2 dG1 (uk ) + dG2 (vj ) + dG2 (vl )

(2)

Trans. Comb. x no. x (2015) xx-xx

≤

2 2 + 2n2

∑

+

≤

n2 1 + n2

∑

ui ,uk ∈E(G1 ) vj ∈V (G2 ) vl ∈V (G2 )

1 { n2 = 1 + n2 2 + 2n2 +

∑

B. S. Shwetha, V. Lokesha and P. S. Ranjini

∑

∑

∑

∑

∑

∑

∑

) 1 1 2 + − dG1 (ui ) dG1 (uk ) dG1 (ui )dG1 (uk )

1 dG1 (ui )dG1 (uk )

∑ (

ui ,uk ∈E(G1 ) vj ∈V (G2 ) vl ∈V (G2 )

1 { n2 1 + n2 2 + 2n2

∑ (

∑

ui ,uk ∈E(G1 ) vj ∈V (G2 ) vl ∈V (G2 )

1 2 + 2n2

n2 1 1 ) 1 ( n2 + + + 2 + 2n2 dG1 (ui ) dG1 (uk ) dG2 (vj ) dG2 (vj )

ui ,uk ∈E(G1 ) vj ∈V (G2 ) vl ∈V (G2 )

∑

5

∑

1 1 )} + dG2 (vj ) dG2 (vj ) √ ∑ dG1 (ui ) + dG1 (uk ) − 2 dG1 (ui )dG1 (uk )

ui ,uk ∈E(G1 ) vj ∈V (G2 ) vl ∈V (G2 )

∑

∑

1 dG1 (ui )dG1 (uk ) ui ,uk ∈E(G1 ) vj ∈V (G2 ) vl ∈V (G2 ) } ∑ ∑ ∑ (1 + 1)

+

n2 1 + n2

+

1 2 + 2n2

√

ui ,uk ∈E(G1 ) vj ∈V (G2 ) vl ∈V (G2 )

1 { (n2 )3 (n2 )3 (n2 )2 m1 } = ABC(G1 ) + R(G1 ) + 1 + n2 2 + 2n2 1 + n2 1 + n2 From equation (1),(2)and (3) we get {n 1 1 H(G1 [G2 ]) ≤ ABC(G2 ) + n1 R(G2 ) + n1 n2 m2 (1 + n2 )2 2 } (n2 )3 ABC(G1 ) + (n2 )3 R(G1 ) + (n2 )2 m1 . + 2

(3)

□ Theorem 2.2. Let G1 and G2 be two connected graphs with order n1 , n2 and size m1 , m2 respectively. Then 1{ H(G1 × G2 ) ≤ n1 ABC(G2 ) + 2n1 R(G2 ) + 2n1 m2 8 } + n2 ABC(G1 ) + 2n2 R(G1 ) + 2n2 m1 . Proof. Let V (G1 ) = {u1 , u2 , ...un1 } and V (G2 ) = {v1 , v2 , ...vn2 } be a set of vertex for G1 and G2 respectively. By the Definition of the cartesian product of two graph one can see that, |E(G1 × G2 )| = |E(G1 )||V (G2 )| + |E(G2 )||V (G1 )|

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B. S. Shwetha, V. Lokesha and P. S. Ranjini

dG1 ×G2 (u, v) = dG1 (u) + dG2 (v) ∑

H(G1 × G2 ) =

(ui ,vj ),(uk ,vl )∈E(G1 ×G2 ),(ui ,vj )̸=(uk ,vl )

∑

=

(ui ,vj ),(ui ,vl )∈E(G1 ×G2 ),vj vl ∈E(G2 )

2 dG1 ×G2 (ui , vj ) + dG1 ×G2 (ui , vl )

∑

+

(ui ,vj ),(uk ,vj )∈E(G1 ×G2 ),ui uk ∈E(G1 )

∑

=

∑

ui ∈V (G1 ) vj ,vl ∈E(G2 )

∑

+

∑

vj ∈V (G2 ) ui ,uk ∈E(G1 )

2 dG1 ×G2 (ui , vj ) + dG1 ×G2 (uk , vl )

2 dG1 ×G2 (ui , vj ) + dG1 ×G2 (uk , vj )

2 dG2 (vj ) + dG2 (vl ) + 2dG1 (ui ) 2 dG1 (ui ) + dG1 (uk ) + 2dG2 (vj )

= B1 + B2

(4)

where B1 , B2 are the sums of the above terms, in order. ∑ ∑ 2 B1 = dG2 (vj ) + dG2 (vl ) + 2dG1 (ui ) ui ∈V (G1 ) vj ,vl ∈E(G2 )

≤ =

1 8

∑

+2

(

ui ∈V (G1 ) vj ,vl ∈E(G2 )

1{ 8

∑

∑

∑

1 2 ) 1 + + dG2 (vj ) dG2 (vl ) dG1 (ui )

(

ui ∈V (G1 ) vj ,vl ∈E(G2 )

∑

∑

ui ∈V (G1 ) vj ,vl ∈E(G2 )

) 1 2 1 + − dG2 (vj ) dG2 (vl ) dG2 (vj )dG2 (vl )

1 dG2 (vj )dG2 (vl )

1 } dG1 (ui ) ui ∈V (G1 ) vj ,vl ∈E(G2 ) √ ∑ dG2 (vl ) + dG2 (vj ) − 2 1{ ∑ ≤ 8 dG2 (vl )dG2 (vj )

+2

∑

∑

ui ∈V (G1 ) vj ,vl ∈E(G2 )

∑

∑

1 dG2 (vj )dG2 (vl ) ui ∈V (G1 ) vj ,vl ∈E(G2 ) } ∑ ∑ +2 1

+2

ui ∈V (G1 ) vj ,vl ∈E(G2 )

√

Trans. Comb. x no. x (2015) xx-xx

=

B. S. Shwetha, V. Lokesha and P. S. Ranjini

} 1{ n1 ABC(G2 ) + 2n1 R(G2 ) + 2n1 m2 8

7

(5)

Similarly we get,

} 1{ n2 ABC(G1 ) + 2n2 R(G1 ) + 2n2 m1 8 From equation (4),(5)and (6) we get 1{ H(G1 × G2 ) ≤ n1 ABC(G2 ) + 2n1 R(G2 ) + 2n1 m2 8 } B1 ≤

(6)

+ n2 ABC(G1 ) + 2n2 R(G1 ) + 2n2 m1 . □ Theorem 2.3. For i ∈ {1, 2}, let Gi be a graph of minimum degree δi , maximum degree ∆i ,order ni and size mi . Then m1 m2 n1 2n1 n2 H(G1 oG2 ) ≥ + + ∆1 + n2 ∆2 + 1 ∆1 + ∆2 + n2 + 1 m2 n1 2n1 n2 m1 + + . H(G1 oG2 ) ≤ δ1 + n2 δ2 + 1 δ1 + δ2 + n2 + 1 Proof. The edges of G1 oG2 are partitioned into three subsets E1 , E2 and E3 as follows E1 = {e ∈ E(G1 oG2 ), e ∈ E(G1 )} E2 = {e ∈ E(G1 oG2 ), e ∈ E(G2i ), i = 1, 2...|V (G1 )|} E3 = {e ∈ E(G1 oG2 ), e = uv, u ∈ V (G2i ), i = 1, 2...|V (G1 )| and v ∈ V (G1 )} and if u is a vertex of G1 oG2 , then { dG1 (u) + |V (G2 )| if u ∈ V (G1 ) dG1 oG2 (u) = dG2 (u) + 1 if u ∈ V (G2 ) Let G1 = (Vi , Ei ), i ∈ {1, 2} and let G1 oG2 = (V, E) we have ∑ 2 H(G1 oG2 ) = dG1 oG2 (u) + dG1 oG2 (v) uv∈E(G1 oG2 )

= Q1 + Q2 + Q3 where Q1 =

∑ uv∈E1

≥

(7)

2 dG1 (u) + n2 + dG1 (v) + n2

2m1 m1 = ∆1 + n2 + ∆1 + n2 ∆1 + n2

(8)

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Trans. Comb. x no. x (2015) xx-xx

∑

Q2 = n1

uv∈E2

≥ Q3 =

B. S. Shwetha, V. Lokesha and P. S. Ranjini

2 dG2 (u) + 1 + dG2 (v) + 1

2n1 m2 n1 m2 = ∆2 + 1 + ∆2 + 1 ∆2 + 1 ∑ uv∈E3 ,u∈V1 andv∈V2

≥

(9)

2 dG1 (u) + n2 + dG2 (v) + 1

2n1 n2 ∆1 + ∆2 + n2 + 1

(10)

From equation (7),(8), (9)and (10) we get H(G1 oG2 ) ≥

m1 m2 n1 2n1 n2 + + ∆1 + n2 ∆2 + 1 ∆1 + ∆2 + n2 + 1

Similarly we deduce the lower bound H(G1 oG2 ) ≤

m1 m2 n1 2n1 n2 + + . δ1 + n2 δ2 + 1 δ1 + δ2 + n2 + 1 □

Theorem 2.4. Let G1 and G2 be two connected graphs with order n1 , n2 and size m1 , m2 respectively. Then H(G1 + G2 ) ≥

2 2 n1 n2 R(G1 ) + R(G2 ) + . m1 + 2n2 + 1 m2 + 2n1 + 1 n1 + n2 − 1

Proof. Let V (G1 ) = {u1 , u2 , ...un1 } and V (G2 ) = {v1 , v2 , ...vn2 } be a set of vertex for G1 and G2 respectively. By the Definition of the join of two graph one can see that, if u is a vertex of G1 + G2 , then { dG1 (u) + |V (G2 )| if u ∈ V (G1 ) dG1 +G2 (u) = dG2 (u) + |V (G1 )| if u ∈ V (G2 ) Therefore, H(G1 + G2 ) =

∑ uv∈E(G1 +G2 )

=

∑

uv∈E(G1 )

+

2 dG1 +G2 (u) + dG1 +G2 (v)

2 + dG1 (u) + n2 + dG1 (v) + n2

∑

u∈V (G1 ),v∈V (G2 )

= A1 + A2 + A3

∑ uv∈E(G2 )

2 dG2 (u) + n1 + dG2 (v) + n1

2 dG1 (u) + n2 + dG2 (v) + n1 (11)

Trans. Comb. x no. x (2015) xx-xx

B. S. Shwetha, V. Lokesha and P. S. Ranjini

where A1 =

∑ uv∈E(G1 )

9

2 dG1 (u) + dG1 (v) + 2n2

Since for each edge uv of G, we have dG (u)+dG (v) ≤ |E(G)|+1 with equality iff every other √ edge of G is adjacent to the edge uv. And 1 ≤ du dv the equality hold iff du = dv = 1. Hence ∑ 2 1 A1 ≥ ×√ |E(G1 )| + 1 + 2n2 du dv uv∈E(G1 )

=

2 R(G1 ) m1 + 1 + 2n2

(12)

Similarly A2 ≥

2 R(G2 ) m2 + 1 + 2n1

(13)

Since for any graph G with n vertices du ≤ n − 1, therefore ∑ 2 A3 = dG1 (u) + n2 + dG2 (v) + n1 u∈V (G1 ),v∈V (G2 )

≥

∑

u∈V (G1 ),v∈V (G2 )

=

2 n1 − 1 + n2 + n2 − 1 + n1

n1 n2 n1 + n2 − 1

(14)

From equation (11),(12), (13)and (14) we get H(G1 + G2 ) ≥

2 n1 n2 2 R(G1 ) + R(G2 ) + . m1 + 2n2 + 1 m2 + 2n1 + 1 n1 + n2 − 1 □

Theorem 2.5. For i ∈ {1, 2}, let Gi be a graph of diameter d(Gi ), order ni and size mi . Then H(G1 ⊕ G2 ) ≥

n22 m1 + n21 m2 − 4m1 m2 . n2 (n1 − d(G1 )) + n1 (n2 − d(G2 )) − 2(n1 − d(G1 ))(n2 − d(G2 ))

Proof. Let V (G1 ) = {u1 , u2 , ...un1 } and V (G2 ) = {v1 , v2 , ...vn2 } be a set of vertex for G1 and G2 respectively. ∑ 2 H(G1 ⊕ G2 ) = dG1 ⊕G2 (ui , vj ) + dG1 ⊕G2 (uk , vl ) (ui ,vj ),(uk ,vl )∈E(G1 ⊕G2 )

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Trans. Comb. x no. x (2015) xx-xx

=

∑

B. S. Shwetha, V. Lokesha and P. S. Ranjini

∑

∑

vj ∈V (G2 ) vl ∈V (G2 ) ui ,uk ∈E(G1 )

+

∑

∑

∑

ui ∈V (G1 ) uk ∈V (G1 ) vj ,vl ∈E(G2 )

−

∑

∑

ui ,uk ∈E(G1 ) vj ,vl ∈E(G1 )

2 dG1 ⊕G2 (ui , vj ) + dG1 ⊕G2 (uk , vl ) 2 dG1 ⊕G2 (ui , vj ) + dG1 ⊕G2 (uk , vl )

2 dG1 ⊕G2 (ui , vj ) + dG1 ⊕G2 (uk , vl )

(15)

As per the definition of the Symmetric difference of a graph dG1 ⊕G2 (ui , vj ) + dG1 ⊕G2 (uk , vl ) = n2 [dG1 (ui ) + dG1 (uk )] + n1 [dG2 (vj ) + dG2 (vl )] − 2[dG1 (ui )dG2 (vj ) + dG1 (uk )dG2 (vl )] Since for each vertex u of a graph G with n vertices, dG (u) ≤ n−eccG (u) and the diameter d(G) ≥ eccG (u). Therefore, dG1 ⊕G2 (ui , vj ) + dG1 ⊕G2 (uk , vl ) ≤ n2 [n1 − eccG1 (ui ) + n1 − eccG1 (uk )] + n1 [n2 − eccG2 (vj ) + n2 − eccG2 (vl )] − 2[(n1 − eccG1 (ui ))(n2 − eccG2 (vj )) + (n1 − eccG1 (uk ))(n2 − eccG2 (vl ))] ≤ n2 [2n1 − 2d(G1 )] + n1 [2n2 − 2d(G2 )] − 2[(n1 − d(G1 ))(n2 − d(G2 )) + (n1 − d(G1 ))(n2 − d(G2 ))] = 2{n2 [n1 − d(G1 )] + n1 [n2 − d(G2 )] − 2[(n1 − d(G1 ))(n2 − d(G2 ))]} (16) From equation (15)and (16) we get H(G1 ⊕ G2 ) ≥

n22 m1 n2 [n1 − d(G1 )] + n1 [n2 − d(G2 )] − 2[(n1 − d(G1 ))(n2 − d(G2 ))]

n21 m2 n2 [n1 − d(G1 )] + n1 [n2 − d(G2 )] − 2[(n1 − d(G1 ))(n2 − d(G2 ))] 4m1 m2 − n2 [n1 − d(G1 )] + n1 [n2 − d(G2 )] − 2[(n1 − d(G1 ))(n2 − d(G2 ))]

+

=

n22 m1 + n21 m2 − 4m1 m2 . n2 [n1 − d(G1 )] + n1 [n2 − d(G2 )] − 2[(n1 − d(G1 ))(n2 − d(G2 ))] □

Trans. Comb. x no. x (2015) xx-xx

B. S. Shwetha, V. Lokesha and P. S. Ranjini

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Acknowledgments The authors wish to record their sincere thanks to the anonymous referees for carefully reading the manuscript and making suggestions that improve the content and presentation of the paper.

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Trans. Comb. x no. x (2015) xx-xx

B. S. Shwetha, V. Lokesha and P. S. Ranjini

B. Shwetha Shetty, Don Bosco Institute of Technology, Bangalore-78, India Email:[email protected] V. Lokesha, Dept. of Mathematics, V S K University, Bellary - 583105, India Email:[email protected] P. S. Ranjini, Department of Mathematics, Don Bosco Institute of Technology, Bangalore-78, India Email:ranjini p [email protected]