The multiplicative version of degree distance and the multiplicative version

J. Math. Comput. Sci. 6 (2016), No. 4, 568-580 ISSN: 1927-5307

THE MULTIPLICATIVE VERSION OF DEGREE DISTANCE AND THE MULTIPLICATIVE VERSION

R. MURUGANANDAM1,∗, R.S. MANIKANDAN2, M. ARUVI3

1_{Department of Mathematics, Government Arts College, Tiruchirappalli 620022, India}

2_{Department of Mathematics, Bharathidasan University Constituent College, Tiruchirappalli, 621601, India} 3_{Department of Mathematics, Anna University, Tiruchirappalli 620024, India}

Copyright c2016 Muruganandam, Manikandan and Aruvi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract. In this paper, the multiplicative version of degree distance and the multiplicative version of Gutman index of Cartesian product of graphs are derived and the indices are evaluated for some well-known graphs such as hypercube, hypertorus and grid.

Keywords:Cartesian product; Multiplicative version of degree distance; Multiplicative version of Gutman index; Degree distance; Wiener index.

2010 AMS Subject Classification:05C07, 05C76.

1. Introduction

Throughout this paper, we consider simple graphs which are finite, indirected graphs without loops and multiple edges. Suppose Gis a graph with a vertex setV(G)and an edge set E(G). For a graphG,the degree of a vertexvis the number of edges incident tovand is denoted by

d_G(v). A topological index Top(G) of a graphGis a number with the property that for every

∗_{Corresponding author}

Received November 26, 2015

graphH is isomorphic toG,Top(H) =Top(G). Notations and definitions which are not given here can be found in[1]and[2]. The Wiener index of a graphGdenoted byW(G)is defined as

W =W(G) =∑{u,v}⊆V(G)dG(u,v) = 1₂∑u,v∈V(G)dG(u,v). In [10],[11], the multiplicative

ver-sion of the Wiener index was conceived by Gutman et al: π =π(G) =∏{u,v}⊆V(G)dG(u,v) =

1

2∏u,v∈V(G)dG(u,v).

The topological indices and graph invariants based on distances between vertices of a graph are widely used for characterizing molecular graphs, establishing relationships between struc-ture and properties of molecules, predicting biological activity of chemical compounds and making their chemical applications[7]. The Wiener index is one of the most topological indices with high correlation with many physical and chemical indices of molecular compounds.

There are some topological indices based on degrees such as the first and second Zagreb indices of molecular graphs. The first and second kinds of Zagreb indices were first introduced in[6](see also[5]). It is reported that these indices are useful in the study of anti-inflammatory activities of certain Chemical instances, and in other practical aspects. The first Zagreb index

M₁(G)and second Zagreb indexM₂(G)of graphGare defined as M₁(G) =∑uv∈E(G)[dG(u) + d_G(v)] =∑u∈V(G)dG2(u)andM2(G) =∑uv∈E(G)dG(u)dG(v).

The degree distance was introduced by Dobrynin and Kochetova [3] and Gutman [4] as a weighted version of the Wiener index. The degree distance ofG, denoted byDD(G),is defined as

DD(G) =

_∑

{u,v}⊆V(G)

d_G(u,v)[d_G(u) +d_G(v)] =1

2

∑

u,v∈V(G)

d_G(u,v)[d_G(u) +d_G(v)]

with the summation going over all pairs of vertices ofG.

In [4], Gutman defined the Schultz index of the second kind which is now known as the Gutman index . The Gutman index ofGdenoted byGut(G),is defined as

Gut(G) = 1

2

∑

u,v∈V(G)

d_G(u,v)[d_G(u)d_G(v)],

the summation runs over all the pairs of vertices ofG. Recently , Todeschini et al[12,13]have proposed the multiplicative variants of ordinary Zagreb indices, which are defined as follows:

∏

=

_∏

(G) =

_∏

v∈V(G)

d2_G(v) =

_∏

uv∈E(G)

and

∏

=

_∏

(G) =

_∏

uv∈E(G)

[d_G(u)d_G(v)].

In this paper, we have found a new graph invariant named multiplicative version of degree distance and multiplicative version of the Gutman index, which can be seen as a weighted version of the Wiener index that isDD∗(G) =1₂∏u,v∈V(G)dG(u,v)[dG(u) +dG(u)],Gut∗(G) =

1

2∏u,v∈V(G)dG(u,v)[dG(u)dG(u)].

The Cartesian product of the graphG₁andG₂,denoted byG₁G₂has the vertex setV(G₁G₂) =V(G1)×V(G2)and(u,x)(v,y)is an edge ofG1G2isu=vandxy∈E(G2)oruv∈E(G1) andx=y.

In this paper, we obtain the multiplicative version of degree distance and the multiplicative version of the Gutman index of the cartesian product of graphs. Also we apply some of our re-sults to compute the exact multiplication version of degree distance and the exact multiplicative version of the Gutman index ofC₄nanotube,C₄nanotorus and grid.

Throughout this paper the degree distance and the Gutman index are denoted asDD+(G)and

Gut+(G). And the multiplicative version of degree distance and the multiplicative version of

the Gutman index are denoted asDD∗(G)andGut∗(G)respectively.

2. Multiplicative version of degree distance of cartesian product of graphs

We begin this section with standard inequality as follows:

Lemma 2.1. (Arithmetic Geometric inequality)Let x1,x2, ...,xnbe non-negative numbers. Then x1+x2+...+xn

n ≥ n

√

x₁x₂...x_n.

In this section, we compute the multiplicative version of the degree distance of the Carte-sian product of G₁G₂ of the graphs G₁ and G₂. Let V(G₁) ={u₀,u₁, ...,u_n1−1},V(G₂) =

{v₀,v₁, ...,v_n2−1},and letw_{i j} denote the vertex(u_i,v_j)ofG₁G₂.

The following lemma follows from the definition of Cartesian product of two graphs.

Lemma 2.2. Let G₁and G₂be two connected graphs. Let w_{i j} = (u_i,v_j)and w_pq = (u_p,v_q)be in V(G₁G₂). Then d_G1G2(w_{i j},w_pq) =d_G1(u_i,u_p) +d_G2(v_j,v_q)and d_G1G2(w_{i j}) =d_G1(u_i) +

Theorem 2.3. If G₁and G₂are two connected graphs with |V(G₁)|=n₁and |V(G₂)|=n₂, where n₁,n₂≥2, then DD∗(G)≤23n1n2−1

h_n

2DD+(G1)+4e(G2)W(G1)

n1n2

in1n2

×hn1DD+(G2)+4e(G1)W(G2)

n1n2

in1n2

×n2(n2−1)DD+(G1)+4(n2−1)e(G2)W(G1)+4(n1−1)e(G1)W(G2)+n1(n1−1)DD+(G2)

n1n2

n1n2

,

where W(G),DD(G) and e(G) denote the Wiener index, degree distance and the number of

edges of G respectively.

Proof.LetG=G1G2.Then

DD∗(G) = 1

2_{wi j,wpq∈V}

∏

= 1 2

n2−1

∏

j=0

n1−1

∏

i,p=0,i6=p

d_G(wi j,wp j)[dG(wi j) +dG(wp j)]

×

n1−1

∏

i=0

n2−1

∏

j,q=0,j6=q

dG(wi j,wiq)[dG(wi j) +dG(wiq)]

×

n2−1

∏

j,q=0,j6=q n1−1

∏

i,p=0,i6=p

dG(wi j,wpq)[dG(wi j) +dG(wpq)]

= 1

2(A1×A2×A3), ...(1)

whereA1,A2 andA3are the products of the above terms, in order. We calculateA1,A2 andA3 of(1)separately.

First we compute

A₁ =

n2−1

∏

j=0

n1−1

∏

i,p=0,i6=p

d_G(w_{i j},w_{p j})d_G(w_{i j}) +d_G(w_{p j})

=

n2−1

∏

j=0

n1−1

∏

i,p=0,i6=p

d_G1(u_i,u_p)

d_G1(u_i) +d_G2(v_j) +d_G1(u_p) +d_G2(v_j)

byLemma(2.2)

=

n2−1

∏

j=0

n1−1

∏

i,p=0,i6=p

d_G1(ui,up)[dG1(ui) +dG1(up) +2dG2(vj)]

≤ ∑

n2−1

j=0 ∑

n1−1

i,p=0,i6=pdG1(ui,up)[dG1(ui) +dG1(up) +2dG2(vj)] n1n2

n1n2

byLemma(2.1)

=

h_∑n_j2₌−₀12DD+(G₁) +4W(G₁)d_G2(v_j) n₁n₂

by the definition of Wiener index and the degree distance of a graph

A₁≤h2n2DD

+_(G

1) +8e(G2)W(G1)

n₁n₂

in1n2

...(2)

Next we compute

A₂ =

n1−1

∏

i=0

n2−1

∏

j,q=0,j6=q

d_G(w_{i j},w_iq)[d_G(w_{i j}) +d_G(w_iq)]

=

n1−1

∏

i=0

n2−1

∏

j,q=0,j6=q

d_G2(v_j,v_q)d_G1(u_i) +d_G2(v_j) +d_G1(u_i) +d_G2(v_j)_byLemma(2.2)

=

n1−1

∏

i=0

n2−1

∏

j,q=0,j6=q

d_G2(v_j,v_q)[2d_G1(u_i) +d_G2(v_j) +d_G2(v_q)]

≤ h∑

n1−1

i=0 ∑

n2−1

j,q=0,i6=qdG2(vj,vq)[2dG1(ui) +dG2(vj) +dG2(vq)] n₁n₂

in1n2

byLemma(2.1)

=

h_∑n1−1 i=0 2DD

+_(G

2) +4W(G2)dG1(ui) n₁n₂

in1n2

by the definition of Wiener index and the degree distance of a graph

A₂≤h2n1DD

+_(G

2) +8e(G1)W(G2)

n1n2

in1n2

...(3)

Next we compute

A3 =

n2−1

∏

j,q=0,j6=q n1−1

∏

i,p=0,i6=p

dG(wi j,wpq)

dG(wi j) +dG(wpq)

=

n2−1

∏

j=0,q=0,j6=q ni−1

∏

i,p=0,i6=p

dG1(ui,up) +dG2(vj,vq)

×hdG1(ui) +dG2(vj) +dGi(up) +dG2(vq) i

≤ h∑

n2−1

j,q=0,j6=q∑ n1−1

i,p=0,i6=p[dG1(ui,up) +dG2(vj,vq)] +

d_G1(u_i) +d_G2(v_j) +d_G1(u_p) +d_G2(v_q)

n1n2

in1n2

= h S

n1n2

in1n2

.

Here

S =

_∑

j,q=0,j6=q h

2DD+(G₁) +2W(G₁)[d_G2(v_j) +d_G2(v_q)] +d_G2(v_j,v_q)4(n₁−1)e(G₁)

where

n1−1

∑

i,p=0,i6=p

1=2

n₁

2

and

n1−1

∑

i,p=0,i6=p

[d_G1(u_i) +d_G1(u_p)] =4(n₁−1)e(G₁)

and by the definition of Wiener index and the additive version of degree distance of a graph.

A₃≤

h2n₂(n₂−1)DD+(G₁) +8(n₂−1)e(G₂)W(G₁) +8(n₁−1)e(G₁)W(G₂) +2n₁(n₁−1)DD+(G₂) n1n2

in1n2

(4)

Using(2),(3)and(4)in(1), we get

DD∗(G)≤ 1₂h2n2DD+(G1)+8e(G2)W(G1)

n1n2

n1n2

×2n1DD+(G2)+8e(G1)W(G2)

n1n2

n1n2

×2n2(n2−1)DD+(G1)+8(n2−1)e(G2)W(G1)+8(n1−1)e(G1)W(G2)+2n1(n1−1)DD+(G2)

n1n2

n1n2i

DD∗(G)≤23n1n2−1 h

n2DD+(G1)+4e(G2)W(G1)

n1n2

in1n2

×hn1DD+(G2)+4e(G1)W(G2)

n1n2

in1n2

×hn2(n2−1)DD+(G1)+4(n2−1)e(G2)W(G1)+4(n1−1)e(G1)W(G2)+n1(n1−1)DD+(G2)

n1n2

in1n2

.

Lemma 2.4. [10]Let P_nand C_ndenote the path and the cycle n on vertices respectively.

(1).For,n≥2,W(P_n) = n(n 2₋₁₎

6

(2).For,n≥3,W(C_n) =



 

 

n3

8, if n is even,

n(n2−1)

8 , if n is odd.

Lemma 2.5. Let P_nand C_ndenote the path and the cycle on n vertices, respectively.

(1)For,n≥2,DD(P_n) =n(n−1)(2n−1) 3

(2)For,n≥3,DD(C_n) =



 

 

n3

2, if n is even,

n(n2−1)

2 , if n is odd.

Corollary 2.6.The graphsR=P_mC_n,S=C_mC_n,n≥3 andm≥2 andT =P_mP_n,m,n≥2 are known asC₄nanotube,C₄nanotorus and grid respectively. (1)DD∗(P_mP_n)

≤ 2

3mn−1 33mn

(m−1)(4mn+n−2m−1) n

mn

×(n−1)(4mn+m−2n−1) m

mn

× (2(n−1)(m−1)((2mm−1)(m+n)−(m−n)

2₎₎

m

mn

2)DD∗(P_mC_n)

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

23mn−1

1

3(m−1)(4m+1)

mn ×n2

2m(2m−1)

mn m₃(n−1)(m−1)(4m+1)+n 2

2(m−1)(2m−1)

m

mn

,

ifnis even.

23mn−1(1₃(m−1)(4m+1))mn(₂1_m(n2−1)(2m−1))mn

×m3(n−1)(m−1)(4m+1)+ 1

2(m−1)(n

2_{−1)(2m−1)}

m

mn

,

ifnis odd.

3).DD(C_mC_n)≤                                                   

23mn−1m2mn×n2mn× m2(n−1) +n2(m−1)mn, ifmis even andnis even.

23mn−1(m2mn)×(n2−1)mn× (n−1)(m2+mn+m−n−1)mn, ifmis even andnis odd.

23mn−1(m2−1)mn×n2mn× (m−1)(n2+mn+n−m−1)mn, ifmis odd andnis even.

23mn−1(m2−1)mn×(n2−1)mn× (n−1)(m−1)(m+n+2)mn, ifmis odd andnis odd.

3. Multiplicative version of Gutman index of Cartesian product of graphs.

Theorem 3.1. If G₁and G₂are two connected graphs with |V(G₁)|=n₁and |V(G₂)|=n₂, where n₁,n₂≥2, then

Gut∗(G₁G₂) ≤ 23n1n2−1hn2Gut

+_(G

1) +2e(G2)DD+(W1)M1(G2)

n1n2

in1n2

× hn1Gut

+_(G

2) +2e(G1)DD+(G2) +W(G1)M1(G2)

n₁n₂

in1n2

× h S1

n₁n₂ in1n2

,

Here

S₁=n₂(n₂−1)Gut+(G₁) +2(n₂−1)e(G₂)DD+(G₁) +W(G₁)(4e(G₂)2−M₁(G₂))

+W(G₂)(4e(G₁)2−M₁(G₁)) +2(n₁−1)e(G₁)DD+(G₂)+n₁(n₁−1)Gut+(G₂),

where Gut(G)+,W(G),M₁(G) and DD+(G) denote the Gutman index, the Wiener index, the

first Zagreb index and the degree distance of G.

Proof.LetG=G1G2. Then

Gut∗(G) = 1

2_{wi j,wpq∈V}

∏

= 1 2{

n2−1

∏

j=0

n1−1

∏

i,p=0,i6=p

d_G(w_{i j},w_{p j})[d_G(w_{i j})d_G(w_{p j})]

×

n1−1

∏

i=0

n2−1

∏

j,q=0,j6=p

d_G(w_{i j},w_iq)[d_G(w_{i j})d_G(w_iq)]

×

n2−1

∏

j,q=0,j6=q n1−1

∏

i,p=0,i6=p

d_G(wi j,wpq)[dG(wi j)dG(wpq)]}

= 1

2(A1×A2×A3), ...(5)

whereA₁,A₂ andA₃are the products of the above terms, in order. We calculateA₁,A₂ andA₃

First we compute

A₁ =

n2−1

∏

j=0

n1−1

∏

i,p=0,i6=p

d_G(w_{i j},w_{p j})[d_G(w_{i j})d_G(w_{p j})]

=

n2−1

∏

j=0

n1−1

∏

i,p=0,i6=p

d_G1(ui,up)(dG1(ui) +dG2(vj))(dG1(up) +dG2(vj))by Lemma(2.2)

≤ h∑

n2−1

j=0 ∑

n1−1

i,p=0,i6=pdG1(ui,up) dG1(ui) +dG2(vj))(dG1(up) +dG2(vj)

n1n2

in1n2

by Lemma(2.1)

= h∑

n2−1

j=0 2Gut+(G1) +2dG2(vj)DD

+_(G

1) +2dG22(vj)W(G1) n1n2

in1n2

by the definitions of the Gutman index, the degree distance and the Wiener index of a graph.

A₁≤h2n2Gut

+_(G

1) +4e(G2)DD+(G1) +2W(G1)M1(G2)

n₁n₂

in1n2

...(6)

Next we computeA₂=∏n_i=1−₀1∏n_j,2_q−₌1_0,j6=qdG(wi j,wiq)dG(wi j)dG(wiq)

=∏n_i=1−₀1∏n_j,2_q−₌1_0,j6=qdG2(vj,vq)[dG1(ui) +dG2(vj)][dG1(ui) +dG2(vq)]byLemma(2.2)

≤h∑

n₁−1

i=0 ∑

n₂−1

j,q=0,i6=qdG2(vj,vq)[2dG1(ui)+dG2(vj)][dG1(ui)]+dG2(vq)

n1n2

in1n2

byLemma(2.1)

=

h_∑n1−1

i=0 2Gut+(G2)+2dG₁(ui)DD+(G2)+2dG2₁(vj)W(G2)

n1n2

in1n2

by the definition of the Gutman index,degree distance and the Wiener index of a graph

A₂≤h2n1Gut+(G2)+4e(G1)DD+(G2)+2W(G2)M1(G1)

n1n2

in1n2

...(7)

Finally we compute

A₃=∏n_j,2_q−₌1_0,j6=q.∏n_i,1_p−₌1_0,i6=pdG(wi j,wpq)[dG(wi j)dG(wpq)]

≤∑ n₂−1

j,q=0,j6=q∑ n₁−1

i,p=0,i6=p[dG1(ui,up)+dG2(vj,vq)][dG1(ui)+dG2(vj)][dG1(up)+dG2(vq)]

n1n2

n1n2

=h S2

n1,n2 in1n2

,

where S2=

n2−1

∑

j,q=0,j6=q.i,q=

∑

d_G1(ui,up)dG1(ui)dG1(up) +dG1(vi,up)dG1(vi)dG2(vq)

+d_G1(ui,up)dG2(vj)dG1(up) +dG1(ui,up)dG2(vj)dG2(vq)

+dG2(vj,vq)dG1(ui)dG1(up) +dG2(vj,vq)dG1(ui)dG2(vq)

+d_G2(v_j,v_q)d_G2(v_j)d_G1(u_p) +d_G2(v_j,v_q)d_G2(v_j)d_G2(v_q)i.

≤ h S3

n₁,n₂ in1n2

Here S₃=

n2−1

∑

j,q=0,j6=q h

2GutG₁+ [d_G2(v_q) +d_G2(v_j)]DD(G₁) +2W(G₁)d_G2(v_j)d_G2(v_q)

+d_G2(v_j,v_q)(4e(G₁)2−M₁(G₁)) +2(n₁−1)d_G2(v_j,v_q)(d_G2(v_j))

+d_G2(v_q))e(G₁) +n₁(n₁−1)d_G2(v_j,v_q)d_G2(v_j)d_G2(v_q)

i

.

by the definitions of the Gutman index, degree distance, the Wiener index and first Zagreb index of a graph.

A₃ ≤ h S4

n1n2

in1n2

, ...(8)

where, S₄=2n₂(n₂−1)Gut(G₁) +4(n₂−1)e(G₂)DDD(G₁) +2W(G₁)(4e(G₂)2−M₁(G₂))

+2W(G₂)(4e(G₁)2−M₁(G₁)) +4(n₁−1)e(G₁)DD(G₂) +2n₁(n₁−1)Gut(G₂).

Since

n2−1

∑

j,q=0,j6=q

Using(6),(7)and(8)in(5), we get

Gut∗(G1G2) ≤ 23n1n2−1

hn₂Gut+(G₁) +2e(G₂)DD+(W₁)M₁(G₂) n₁n₂

in1n2

× hn1Gut

+_(G

2) +2e(G1)DD+(G2) +W(G1)M1(G2)

n1n2

in1n2

× h S1

n1n2

in1n2

,

S₁ = n₂(n₂−1)Gut+(G₁) +2(n₂−1)e(G₂)DD+(G₁) +W(G₁)(4e(G₂)2−M₁(G₂))

+ W(G₂)(4e(G₁)2−M₁(G₁)) +2(n₁−1)e(G₁)DD+(G₂)+n₁(n₁−1)Gut+(G₂).

This completes the proof.

Lemma 3.2. Let Pnand Cndenote the path and the cycle on nvertices, respectively.

(1)For,n≥2,Gut(P_n) = (n−1)(2n₃2−4n+3)

(2)For,n≥3,Gut(C_n) =



 

 

n3

2, if n is even

n(n2−1)

2 , if n is odd

(3)For,n≥2,M₁(P_n) =4n−6andM₁(P₁) =0

(4)For,n≥3,M1(Cn) =4n Using Theorem3.1, Lemmas 2.4 2.5and3.2, we obtain the exact multiplication version of Gutman index of the following graphs.

Corollary 3.3.The graphs R=P_mC_n,S=C_mC_n,n≥3and m≥2and T =P_mP_n,m,n≥2

are known as C4nanotube,C4nanotorus and grid respectively. 1) Gut(PmPn)

≤ 23mn−1× (m−1)

mn ((2mn−m)(m−1) +n) mn

× (n−1)

mn ((2mn−n)(n−1) +m) mn

× (n−1)(m−1)

3mn (6mn+5)(m+n)−4(m

2_+n2_+3mn)

+ 1

6mn(4−2mn)(m

2

2) Gut∗(P_mC_n)

≤ 23mn−1× (m−1)

3mn ((2m)(2m+1)(m−2) +6mn(m−1)−3) mn

× n

2

4m(8m−7) mn

× (m−1)

3mn

3n(n−1) +2m3(2m−1)−2m(m−1) +4m3n(n−1)

+ 2mn(m−n)−4mn(mn−1)+ n 2 4m(6m

2_−8m+2n2₊₇₎mn

,i f niseven

Gut∗(P_mC_n) ≤ 23mn−1× (m−1)

3mn ((2m)(2m+1)(m−2) +6mn(m−1)−3)

mn n2−1

4m (8m−7) mn

× (m−1)

3mn 3n(n−1) +2m

3_{(2m−1)−2m(m−1) +4m}3_n(n−1)

+ 2mn(m−n)−4mn(mn−1)+ n 2₋₁

4m (6m

2_−8m+2n2₊₇₎mn

,i f nisodd

3) Gut∗(CmCn)≤24mn−1m2mnn2mn h

3mn(m+n)−(m2+n2)+m3(m2−n)+_mnn3(n2−m)imn,if m is even and n is even

Gut∗(C_mC_n)≤24mn−1m2mn(n2−1)2mn

h

3(n−1)

m2+(n+1)(m−1)

+

m3(m2−n)+(n3−mn)(n2−1)

mn

imn ,

if m is even and n is odd

Gut∗(C_mC_n)≤24mn−1(m2−1)mnn2mnh3(m−1) n2+(n−1)(m+1)+(m3−mn)(m2_mn−1)+n3(n2−m)imn, if m is odd and n is even

Gut∗(C_mC_n)≤24mn−1(m2−1)mn(n2−1)mnh3

(m2−1)(n−1) + (n2−1)(m−1) +(m3−mn)(m2−1_mn)+(n3−mn(n2−1))imn, if m is odd and n is odd.

Conflict of Interests

The authors declare that there is no conflict of interests.

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