You Zhi-fuand Huang Yu-fei

(1.School of Computer Science,Guangdong Polytechnic Normal University,Guangzhou,510665)

(2.Department of Mathematics Teaching,Guangzhou Civil Aviation College,Guangzhou,510403)

Communicated by Du Xian-kun

Abstract:In this note,we correct a wrong result in a paper of Das et al.with regard to the comparison between the Wiener index and the Zagreb indices for trees(Das K C,Jeon H,Trinajsti′c N.The comparison between the Wiener index and the Zagreb indices and the eccentric connectivity index for trees.Discrete Appl.Math.,2014,171:35–41),and give a simple way to compare the Wiener index and the Zagreb indices for trees.Moreover,the comparison between the Wiener index and the Zagreb indices for unicyclic graphs is carried out.

Key words:Wiener index,Zagreb indice,tree,unicyclic graph

1 Introduction

Throughout this paper,let G be a simple connected graph with vertex set V(G)and edge set E(G).The order and size of G are defined as n=|V(G)|and m=|E(G)|,respectively.For a simple connected graph G,if m=n?1,then G is called a tree;if m=n,then G is called a unicyclic graph.The degree of a vertex vi∈V(G)in G is denoted by dG(vi).The distance between two vertices vi,vj∈V(G)is the length of the shortest path between viand vj,denoted by dG(vi,vj).

Molecular descriptors play an important role in mathematical chemistry,especially in the QSPR and QSAR modeling.Among them,a special place is reserved for the so called topological indices.Nowadays,there exists a legion of topological indices that found applications in various areas of chemistry(see[1]).Among the oldest and most studied topological indices,there are two classical vertex-degree based topological indices:the first Zagreb index M1(G)and the second Zagreb index M2(G),which are defined,respectively,as

Many works on the Zagreb indices have been proposed(see[1]and[2]and the references cited therein).Moreover,one of the oldest and most thoroughly studied distance based on molecular structure descriptors is the Wiener index(see[3]and[4]):

For details on the Wiener index see the review[5]and the references cited therein.

Recently,Das et al.[6]compared the Wiener index and the Zagreb indices for trees.However,we found that one of the main results in[6]was incorrect.

In this note,we correct the wrong result in[6]and give a simple way to compare the Wiener index and the Zagreb indices for trees.Besides,the comparison between the Wiener index and the Zagreb indices for unicyclic graphs is carried out.

2 Comparison Between the Wiener Index and the Zagreb Indices for Trees

Error 2.1([6],Corollary 2.3) Let T be a tree of order n(n>3).Then W(T)≥M1(T).

As usual,we denote by K1,n?1(or Sn)the star of order n(n ≥ 2),Pnthe path of order n(n≥2),and Cnthe cycle of order n(n≥3),respectively.Denote by DSp,q(p≥q≥1,n=p+q+2),a double star of order n(n≥4)which is constructed by joining the central vertices of two stars K1,pand K1,q.Other notations and terminology are not defined here which will conform to those in[7].

Example 2.1For the star Snof order n(n≥2),

It can be seen that the star Sn(n≥2)is a counter example for Corollary 2.3 in[6].

Lemma 2.1[8],[9]Let n ≥ 4,and T be a tree of order n.If T?Sn,DSn?3,1(see Fig.2.1),then M1(T)≤ M1(DSn?3,1)

Fig.2.1 DSn?3,1

Lemma 2.2[5]Let n≥5,and T be a tree of order n.If T?Sn,Pn,then

Lemma 2.3Let n≥5,and T be a tree of order n.If T?Sn,then

Proof.By Lemma 2.2,one has

By Lemma 2.1,we get

Noting that

we have

The result holds.

Lemma 2.4Let T be a tree of order n(2≤n≤4).Then

with equality holding if and only if T～=P4.

Proof. There are four trees of order n(2≤n≤4):S2,S3,S4,P4.By Example 2.1,

Besides,by direct calculation,one has

This completes the proof.

By Example 2.1,Lemmas 2.3 and 2.4,the following theorem holds.

Theorem 2.1Let T be a tree of order n(n≥2).Then

(1)W(T)

(2)W(T)=M1(T)if and only if T～=P4;

(3)W(T)>M1(T)if and only if T?Sn,P4.

It can be seen that we correct the wrong result in[6](Error 2.1)and give a simple way to compare the Wiener index and the first Zagreb index for trees.Besides,we also give a simple way to prove the Theorem 2.4 in[6]by using a similar method.

Theorem 2.2[6]Let T be a tree of order n(n≥2).Then W(T)≥M2(T)with equality holding if and only if T～=Sn.

Proof.By Lemmas 2.1 and 2.2,we have

the equality holds if and only if T～=Sn.

3 Comparison Between the Wiener Index and the Zagreb Indices for Unicyclic Graphs

Lemma 3.1[9]Let n≥7,and U1,U2be unicyclic graphs of order n.

and the third inequality holds when n≥10.

Lemma 3.2[10]Let n≥6,and U be a unicyclic graph of order n.Ifthen

By directly computing the Zagreb indices and the Wiener index of the unicyclic graphshave the results shown as Table 3.1.

Table 3.1 The Zagreb indices and the Wiener index of some unicyclic graphs

Lemma 3.3Let n≥7,and U be a unicyclic graph of order n.Then

Proof.By Lemmas 3.1 and 3.2 we have

The result holds.

Moreover,by Table 3.1,it can be directly compared the Zagreb indices with the Wiener index for the unicyclic graphsand we get the results shown as Table 3.2.

Table 3.2 Comparison between the Wiener index and the first Zagreb index for the unicyclic

Table 3.2 Comparison between the Wiener index and the first Zagreb index for the unicyclic

U W(U)M1(U)G(n)3,1(n≥3) n≥3 – –G(n)3,2(n≥5) 5≤n≤8 n=9 n≥10 G′(n)3,1(n≥5) 5≤n≤7 – n≥8 G(n)4,1(n≥4) 4≤n≤7 n=8 n≥9 G(n)3,3(n≥7) n=7 – n≥8 G(n)3,4(n≥6) n=6 – n≥7

Let U3,U4,U5,U6,U7,U8,U9,U10be the unicyclic graphs depicted in Fig.3.2.

Fig.3.2 Some unicyclic graphs of order n=6

Lemma 3.4Let U be a unicyclic graph of order n=6.Then

Proof. By[11],there are 13 unicyclic graphs of order n=6,that is,Combining the results of Table 3.2 with direct calculations,we have

This completes the proof of Lemma 3.4.

Lemma 3.5 Let U be a unicyclic graph of order n(3≤n≤5).Then W(U)

The result holds.

Combining Lemmas 3.3,3.4 and 3.5 with Table 3.2,we have the following theorem.

Theorem 3.1Let U be a unicyclic graph of order n(n≥3).Then

Finally,the comparison between the Wiener index and the second Zagreb indices for unicyclic graphs is present.

Theorem 3.2 Let n≥15,and U be a unicyclic graph of order n.Then

Proof. (1)By Table 3.1 and directly computing,we obtain the desired results as follows: