Reverse Zagreb indices of corona product of graphs

(1)

https://doi.org/10.26637/MJM0604/0003

Reverse Zagreb indices of corona product of graphs

P. Kandan

1

and A. Joseph Kennedy

2

*

Abstract

In this paper, some exact expressions for the first reverse Zagreb alpha index, first reverse Zagreb beta index and second reverse Zagreb index of corona product of two simple graphs are determined. Using this results we obtained these indices fort−thorny graph and bottleneck graph.

Keywords

Reverse Zagreb index, Corona product,t−thorny graph, bottleneck graph.

AMS Subject Classification

05C12, 05C76.

1_{Department of Mathematics, Annamalai University, Annamalainagar, Annamalainagar-608002, India.} 2_{Department of Mathematics, Pondicherry University, Pondicherry-605014, India.}

*Corresponding author:2_{kennedy.pondi@gmail.com}

Article History: Received10June2018; Accepted16October2018 c2018 MJM.

1. Introduction

LetG= (V(G),E(G))be graph with vertex setV(G)and edge setE(G).All the graphs considered in this paper are sim-ple and connected. Graph theory has successfully provided chemists with a variety of useful tools [1,8,10,11], among which are the topological indices. In theoretical chemistry, assigning a numerical value to the molecular structure that will closely correlate with the physical quantities and activi-ties. Molecular structure descriptors (also called topological indices) are used for modeling physicochemical, pharmaco-logic, toxicopharmaco-logic, biological and other properties of chemical compounds. Zagreb indices were introduced more than forty years ago by Gutman and Trinajestic [9]. The Zagreb indices are found to have applications in QSPR and QSAR studies as well, see [3]. One of the recently introduced indices called Reverse Zagreb indicesby Ediz [5] and he obtained the max-imum and minmax-imum graphs with respect to the first reverse Zagreb alpha index and minimum graphs with respect to the first reverse Zagreb beta index and the second reverse Za-greb index. Kulli [15,16] defined first and second reverse

hyper-Zagreb indices of a graph and obtained first two reverse Zagreb indices, the first two reverse hyper Zagreb indices and their polynomials of rhombus silicate networks, moreover he introduced the product connectivity reverse index of a graphG and obtained it for silicate networks and hexagonal networks.

2. Preliminaries

In this section, we recall some definitions and basic results of Zagreb index of graph which will be used throughout the paper.

Definition 2.1. The first Zagreb index is defined as M1(G) =

∑

u∈V(G)

dG(u)2,where dG(u)denote the degree of the vertex

u in G. In fact, one can rewrite the first Zagreb index as M1(G) = ∑

uv∈E(G)

(dG(u) +dG(v)).

Definition 2.2. The second Zagreb index is defined as M2(G) =

∑

uv∈E(G)

dG(u)dG(v),where dG(u), dG(v)denotes the degree of

the vertex u,v respectively in G.

Definition 2.3. Let G be a simple connected graph. Then the total reverse vertex degree of G defined as T R(G) = ∑

a∈V(G) ca,

where ca(G) = ∆G−dG(a) +1,known as reverse vertex

de-gree of a∈V(G),in shortly ca(G) =caand∆Gdenote the

largest of all degrees of vertices of G.

Definition 2.4. Let G be a simple connected graph. Then the first reverse Zagreb alpha index of G defined as CMα

1(G) =

∑

(2)

Definition 2.5. Let G be a simple connected graph. Then the first reverse Zagreb beta index of G defined as CM₁β(G) =

∑

ab∈E(G)

(ca+cb).

Definition 2.6. Let G be a simple connected graph. Then the second reverse Zagreb index of G defined as CM2(G) =

∑

ab∈E(G) cacb.

In [4], the chemical applications of these new indices have been investigated. In [6], Ediz and Cancan obtained the reverse Zagreb indices for Cartesian product of two simple connected graphs. In this extend we obtain the value of this index for corona product of graphs.

Graph operations play an important role in the study of graph decompositions into isomorphic subgraphs. Thecorona of two graphs was first introduced by Frucht and Harary in [7]. LetGandHbe two simple graphs. Thecorona prod-uct G◦H,is obtained by taking one copy ofGand|V(G)| copies ofH; and by joining each vertex of thei-th copy of Hto thei-th vertex ofG,where 1≤i≤ |V(G)|,see Figure 1. Let the vertex set ofGandHdenoted as {v1,v2,· · ·,vn1}

and {u1,u2,· · ·,un2} respectively. For 1≤i≤n1,denoteH

i

theith copy ofH joined to the vertexvi and let the vertex

set ofHiis {xi1,xi2,· · ·,xin2}, denoted byV2=V(H

i₎_{and let}

V1=V(G0),be the vertex set ofG0,whereG0is a copy ofG

which contained inG◦H.Coronas sometimes appear in chem-ical literature as plerographs of the usual hydrogen-suppressed molecular graphs known as kenographs, see [13] for more in-formation. Different topological indices such as Wiener-type indices [2], Szeged, vertex PI, first and second Zagreb in-dices [18], weighted PI index [14], etc. of the corona product of two graphs have already been studied. For convenience, we partition the edge set ofG◦H into three sets as follow, E1={vivj|vi,vj ∈V(G0)}, E2={xirxis|xir,xis ∈V(Hi)}

andE3={vixir|vi∈V(G0),xir∈V(Hi)},where 1≤i,j≤n1

and 1≤r,s≤n2.From the structure of the corona product

GandH(see Fig.1), one can easily observe the following lemma and corollary.

Lemma 2.7. Let G and H be two connected graphs with n1

and n2vertices, respectively. Then

(a)|V(G◦H)| =|V(G0)|+|V(Hi)|=n1+n1n2

(b)|E(G◦H)|=|E(G0)|+|V(G)||E(H)|+|V(G)||V(H)|=

E1+n1|E(H)|+n1n2

(c) dG◦H(a) =

(

dG(a) +n2, if a∈V1

dH(a) +1, if a∈V2.

Corollary 2.8. Let G and H be two connected graphs with n1

and n2vertices, respectively. Then the∆G◦H =∆G+O(H) =

∆G+n2.

Proof. Let∆Gand∆Hbe the largest degree of the respective

graph. Since∆H<n2,then one can easily observed from the

structure of corona product ofG◦Hand from lemma 2.7 that ∆G◦H =∆G+n2.

3. Main Results

In this section, we have obtain the reverse zagreb index of corona two connected graphs using it we have deduced for some standard graphs .

Proposition 3.1. Let G and H be two connected graphs with n1and n2vertices, respectively. Then

cG◦H(a) =

(

ca(G) if a∈V1

∆(G◦H)−dH(a) +1 if a∈V2

(3.1)

Proof. By using lemma2.7and corollary2.8, we have

cG◦H(a) = ∆G◦H−dG◦H(a) +1

=

(

∆G+n2−(dG(a) +n2) +1 ifa∈V1

∆G+n2−(dH(a) +1) +1 ifa∈V2

=

(

∆G−dG(a) +1 ifa∈V1

∆G+n2−dH(a) ifa∈V2

=

(

ca(G) ifa∈V1

∆(G◦H)−dH(a) +1 ifa∈V2

.

Theorem 3.2. Let G and H be two connected graphs with n1,n2vertices, respectively. Then CM1α(G◦H) =CM1α(G) +

n1

n2∆2_G◦H−4∆G◦HE(H) +M1(H)

.

Proof. Using Proposition 2.3, we have

CMα

1(G◦H) =

∑

(3)

=

_∑

a∈V1

c2_a+

_∑

a∈V2

c2_a

=

_∑

a∈V1

∆G◦H−dG0(a) +1

2

+

_∑

a∈V2

∆G◦H−dHi(a) +1 2

=

_∑

a∈V(G)

(∆G+n2)−(dG(a) +n2) +1

2

+n1

∑

a∈V(H)

(∆_G+n2)−(dH(a) +1) +1

2

=

_∑

a∈V(G)

∆G−dG(a) +1

2

(3.2)

+n1

_∑

a∈V(H)

(∆G+n2)−dH(a)

2

=

_∑

a∈V(G)

ca2+n1

∑

a∈V(H)

(∆G+n2)2

−2

_∑

a∈V(H)

(∆G+n2)dH(a) +

∑

a∈V(H)

d2_H(a)

= CMα

1(G) +n1

n2∆2_G◦H−2∆G◦H(2E(H)) +M1(H)

= CMα

1(G) +n1

n2∆2_G◦H−4∆G◦HE(H) +M1(H)

.

Proposition 3.3. Let Pnbe a path and Knbe a n independent

vertices, with n≥3.Then (a) CMα

1(Pn) =n+6,CM1α(Cn) =n and CM1α(Kn) =0

(b) CM₁β(Pn) =2n,CM1β(Cn) =2n and CM1β(Kn) =0

(c) CM2(Pn) =n+1,CM2(Cn) =n and CM2(Kn) =0.

Lemma 3.4. Let Pnbe a path and Cnbe a cycle with n≥3.

Then T R(Pn) =n+2and T R(Cn) =n.

Thet−thorny graphof a given graphGis obtained as G◦Kt,whereKtdenotes the empty graph ontvertices [12].

Using Theorem 3.2and Proposition 3.3, for thet−thorny graph of a graphG,we have the following corollaries.

Corollary 3.5. Let G be a simple connected graph of order n and size E(G). Then CMα

1(G◦Kt) =CM1α(G) +n1t(∆G+

t)2.

Now by using Corollary 3.5 and Proposition 3.3, we present two formulas for the reverse Zagreb indices of the t−thorny path Pn◦Ktand thet−thorny cycle Cn◦Kt.

Corollary 3.6. Let G be a simple connected graph of order n and size E(G). Then CMα

1(G◦Kt) =CM1α(G) +n1t(∆G+

t)2.

Now by using Corollary3.6and Proposition3.3, we present two formulas for the reverse Zagreb indices of thet−thorny path Pn◦Ktand thet−thorny cycle Cn◦Kt.

Corollary 3.7. For n≥3,if G=Pnand H =Kt, we have

CMα

1(Pn◦Kt) = (n+6) +nt(2+t)2.

Corollary 3.8. For n≥3,if G =Cnand H =Kt, we have

CMα

1(Cn◦Kt) =n(t(2+t)2+1).

Interesting classes of graphs can also be obtained by spe-cializing the first component in the corona product. For ex-ample, for a graph G, the graphG◦K2is called itsbottleneck

graph. For the bottleneck graph of a graph G, we obtain the following corollaries.

Corollary 3.9. Let G be a simple connected graph of order n≥3.Then CMα

1(G◦K2) =CM1α(G) +18n.

Using Corollary3.9and Proposition3.3, the formulas for the bottleneck graph of a cycleCn◦K2 and the bottleneck

graph of a pathPn◦K2are easily obtained.

Corollary 3.10. For n≥3,if G=Cn, we have CM1α(Cn◦

K2) =19n.

Corollary 3.11. For n≥3,if G = Pn, we have CM1α(Pn◦

K2) =19n+6.

Theorem 3.12. For a connected graph G and H with n1,n2

vertices respectively, then CM₁β(G◦H) =CM₁β(G) +n1

2∆G◦HE(H)−M1(H)

+n1n2∆G◦H−2n1E(H)+n2T R(G).

Proof. Using Proposition 2.3, we haveCM₁β(G◦H) =∑ab∈E1(ca+cb) +∑ab∈E2(ca+cb) +∑ab∈E3(ca+cb).Let

us calculate each summation separately, for the edge setE1,

we have∑ab∈E1(ca+cb)

=

_∑

ab∈E1

(∆G◦H−dG0(a) +1) + (∆G◦H−dG0(b) +1)

=

_∑

ab∈E(G)

(∆G+n2−(dG(a)+n2)+1)

+(∆G+n2−(dG(b)+n2)+1)

=

_∑

ab∈E(G)

(∆G−dG(a) +1) + (∆G−dG(b) +1)

=

_∑

ab∈E(G)

(ca+cb)

= CM₁β(G).

Now for the edge setE2,we have∑ab∈E2(ca+cb)

=

_∑

ab∈E2

(∆G◦H−dHi(a) +1) + (∆G◦H−dHi(b) +1)

= n1

_∑

ab∈E(H)

(∆G+n2−(dH(a)+1)+1)

+(∆G+n2−(dH(b)+1)+1)

= n1

∑

ab∈E(H)

2(∆G◦H)−(dH(a) +dH(b))

= n1

2∆G◦HE(H)−M1(H)

.

Finally for the edge setE3,we have∑ab∈E3(ca+cb)

= ∑

ab∈E3

a ∈Hi, b∈G0

(∆G◦H−dHi(a) +1) + (∆G◦H−dG0(b) +1)

(4)

= ∑

ab∈E3

a ∈V(H), b∈V(G)

(∆G+n2−(dH(a)+1)+1) +(∆G+n2−(dG(b)+n2)+1)

= ∑

ab∈E3

a ∈V(H), b∈V(G)

(∆G+n2−dH(a)) + (∆G−dG(b) +1)

=

_∑

ab∈E3

a ∈V(H), b∈V(G)

(∆_G+n2−dH(a)) +

∑

ab∈E3

a ∈V(H), b∈V(G)

(∆G−dG(b) +1)

=

_∑

ab∈E3

a ∈V(H), b∈V(G)

(∆G+n2)−

∑

ab∈E3

a ∈V(H), b∈V(G)

dH(a) +n2

∑

ab∈E3

b∈V(G)

(∆G−dG(b) +1)

= n1n2∆G◦H−n1(2E(H)) +n2

∑

b∈V(G) cb

= n1n2∆G◦H−2n1E(H) +n2T R(G).

Hence, we haveCM₁β(G◦H) =CM₁β(G)+n1

2∆G◦HE(H)−

M1(H)

+n1n2∆G◦H−2n1E(H) +n2T R(G).

Using Theorem3.12, Proposition3.3and Lemma3.4, we have the following corollaries

Corollary 3.13. For n≥3,if G=Pnand H =K2, we have

CM₁β(Pn◦K2) =4(4n+1).

CM₁β(Pn◦Kt) =n(t2+3t+2) +2t.

Corollary 3.15. For n≥3,if G=Cnand H =Kt, we have

CM₁β(Cn◦Kt) = n(t2+3t+2).

Theorem 3.16. For a connected graph G and H with n1,n2

vertices respectively, then CM2(G◦H) =CM2(G)+n1

∆2_G_◦_HE(H)− ∆G◦HM1(H) +M2(H)

+n2∆G◦H−2E(H)

T R(G).

Proof. Using Proposition3.1, we have

CM2(G◦H) =

_∑

ab∈EG◦H (cacb)

=

_∑

ab∈E1

(cacb) +

∑

ab∈E2

(cacb) +

∑

ab∈E3

(cacb).

Let us calculate each summation separately, for the edge

setE1we have∑ab∈E1(cacb)

=

_∑

ab∈E1

(∆G◦H−dG0(a) +1)(∆_G_◦_H−d_G0(b) +1)

=

_∑

ab∈E(G)

(∆G+n2−(dG(a)+n2)+1)

(∆G+n2−(dG(b)+n2)+1)

=

_∑

ab∈E(G)

(∆G−dG(a) +1)(∆G−dG(b) +1)

=

_∑

ab∈E(G) (cacb)

= CM2(G).

Now for the edge setE2,we have∑ab∈E2(cacb)

=

_∑

ab∈E2

(∆G◦H−dHi(a) +1)(∆G◦H−dHi(b) +1)

= n1

_∑

ab∈E(H)

(∆G+n2−(dH(a)+1)+1) (∆G+n2−(dH(b)+1)+1)

= n1

_∑

ab∈E(H)

(∆G◦H−dH(a))(∆G◦H)−dH(b))

= n1

∑

ab∈E(H)

∆2G◦H−∆G◦H(dH(a) +dH(b)) +dH(a)dH(b)

= n1

∑

ab∈E(H)

∆2_G◦H−∆G◦H

∑

ab∈E(H)

(dH(a) +dH(b)) +

∑

ab∈E(H)

dH(a)dH(b)

= n1

∆2_G◦HE(H)−∆G◦HM1(H) +M2(H)

.

Finally for the edge setE3,we have∑ab∈E3(cacb)

=

_∑

ab∈E3

a ∈Hi, b∈G0

(∆G◦H−dHi(a) +1)(∆G◦H−dG0(b) +1)

=

_∑

ab∈E3

a ∈V(H), b∈V(G)

(∆G+n2−(dH(a)+1)+1) (∆G+n2−(dG(b)+n2)+1)

=

_∑

ab∈E3

a ∈V(H), b∈V(G)

(∆G+n2−dH(a))(∆G−dG(b) +1)

=

_∑

ab∈E3

a ∈V(H), b∈V(G)

(∆G+n2−dH(a))(cb)

=

_∑

ab∈E3

a ∈V(H), b∈V(G)

cb∆G◦H−

∑

ab∈E3

a ∈V(H), b∈V(G)

dH(a)cb

= n2∆G◦H

∑

b∈V(G)

cb−

∑

a∈V(H)

dH(a)

_∑

b∈V(G)

cb

= n2∆G◦H−

_∑

a∈V(H)

dH(a)

∑

b∈V(G) cb

= n2∆G◦H−2E(H)

(5)

Hence, we haveCM2(G◦H) =CM2(G) +n1

∆2_G_◦_HE(H)− ∆G◦HM1(H) +M2(H)

+n2∆G◦H−2E(H)

T R(G).

As a direct consequence of Theorem3.16and using Propo-sition3.3. we have the following corollaries

Corollary 3.17. For n≥3,if G=Pnand H =K2, we have

CM2(Pn◦K2) =16n+3.

CM2(Pn◦Kt) = (n+1) + (n+2)(t+2)t.

Corollary 3.19. For n≥3,if G=Cnand H =Kt, we have

CM2(Cn◦Kt) =n(t+1)2.

4. Conclusion

In this paper, we have obtained the exact value of the first reverse Zagreb alpha index, first reverse Zagreb beta index and second reverse Zagreb index of corona product of two simple graphs and we have derived these indices fort-thorny graph and bottleneck graph of path and cycle. It would be interesting to study mathematical properties of these modified indices and report their chemical relevance and formulas for some important graph classes. In particular, some exact expressions for the first reverse Zagreb alpha index, first reverse Zagreb beta index and second reverse Zagreb index of other graph operations (such as the composition, join, disjunction and symmetric difference of graphs, bridge graphs and Kronecker product of graphs) can be derived similarly.

Acknowledgment

We thank the referee for the valuable suggestions for the improvement of this paper.

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ISSN(O):2321−5666

Reverse Zagreb indices of corona product of graphs