Extremal Values on Distance-Degree-Based Topological Indices of Cacti with r Cycles

WANG Yu-xi, CHEN Han-lin, DENG Han-yuan

(College of Mathematics and Computer Science, Hunan Normal University, Changsha 410081, China)



Extremal Values on Distance-Degree-Based Topological Indices of Cacti with r Cycles

WANG Yu-xi, CHEN Han-lin, DENG Han-yuan

(College of Mathematics and Computer Science, Hunan Normal University, Changsha 410081, China)

Keywordscactus;thedegreedistanceindex;theGutmanindex;theadditivelyweightedHararyindex;themultiplicativelyweightedHararyindex;extremalvalue

In this paper, we will consider the degree distance index [4,6,19], the Gutman index [2,12], the additively weighted Harary index [1] and the multiplicatively weighted Harary index [17]. They are defined for a connected graphGas follows, repectively

WewilldeterminethemaximalvaluesoftheadditivelyweightedHararyindex,themultiplicativelyweightedHararyindexandtheminimalvalueofthedegreedistanceindex,theGutmanindexamongallcactiofordernwithrcycles,andcharacterizethecorrespondingextremalgraphs.

1Preliminaries

For a graphG, the first Zagreb indexM1(G) and the second Zagreb indexM2(G) ofGare, respectively

ThefirstZagrebcoindexandthesecondZagrebcoindexofGare,respectively

WenowintroducethefollowingnewinvariantsforanygraphG

NotethatM*(G)=2M2(G)-M1(G), N*(G)=M1(G)+M2(G).

Beforestatingourmainresults,wewilllistsomelemmasaspreliminaries,whichwillplayanimportantroleinthenextproofs.

IfG∈G(n,r), thenm=|E(G)|=n+r-1. The following lemma is obtained from Propositions 2 and 4 in [3].

Lemma1[3]IfG∈G(n,r),n≥2, then

Lemma 2(i) LetG∈G(n,r)withlargestM1(G).ThenthemaximumdegreeΔ(G)=n-1.

(iii)LetG∈G(n,r)withlargestN*(G).ThenΔ(G)=n-1.

2Main results

Inthissection,wewillcharacterizethemaximalgraphswithrespecttothemaximalvaluesoftheadditivelyweightedHararyindex,themultiplicativelyweightedHararyindexandtheminimalgraphwithrespecttothedegreedistanceindex,theGutmanindexamongallcactiofordernwithrcycles.

Theorem 3LetG∈G(n,r).Then

(i) M1(G)≤n2-n+6rwithequalityifandonlyifG≅G0(n,r).

(iii) N*(G)≤2n2+2nr-3n+8r+1withequalityifandonlyifG≅G0(n,r).

Now,westudythemaximalvaluesoftheadditivelyweightedHararyindex,themultiplicativelyweightedHararyindexandtheminimalvalueofthedegreedistanceindex,theGutmanindexamongallcactiofordernwithrcycles.

ProofBythedefinitionofHA,wehave

with equality if and only ifG≅G0(n,r).

Theorem5IfG∈G(n,r), thenDD(G)≥(3n+4r- 4)(n-1)-6rwith equality if and only ifG≅G0(n,r).

ProofBythedefinitionoftheDDindex,wehave

M1(G)+4(n+r-1)(n-1)-2M1(G) (by Lemma 1(1))=

4(n+r-1)(n-1)+M1(G)≥

(3n+4r-4)(n-1)-6r(by Theorem 3(i))

with equality if and only ifG≅G0(n,r).

ProofBythedefinitionofHM,wehave

Theorem7IfG∈G(n,r), then Gut(G)≥2n2+6nr-5n-4r2-16r+3 with equality if and only ifG≅G0(n,r).

ProofBythedefinitionoftheGutmanindex,wehave

M2(G)+4(n+r-1)2-2M2(G)-M1(G) (by Lemma 1(2))=

4(n+r-1)2-(M1(G)+M2(G))=

4(n+r-1)2-N*(G) (by the difinition ofN*(G))≥

2n2+6nr-5n+4r2-16r+3 (by Theorem 3(iii))

with equality if and only ifG≅G0(n,r).

By selectingr=0 orr=1 in Theorems 4-7, we can get the following results.

Corollary8Amongalltreesofordern,

(ii)theuniquetreewiththeminimaldegreedistanceindexDDisthestarSn,andDD(Sn)=3n2-7n+4[15];

(iv)theuniquetreewiththeminimalGutmanindexGutisthestarSn,andGut(Sn)=2n2-5n+3[2].

Corollary9Amongallunicyclegraphsofordern,

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(编辑HWJ)

DOI:10.7612/j.issn.1000-2537.2016.04.013

收稿日期：2015-07-31

基金项目：国家自然科学 项目(61572190)；湖南省研究生创新 项目(CX2015B162)

*通讯作者,E-mail：hydeng@hunnu.edu.cn

中图分类号O157.5

文献标识码A

文章编号1000-2537(2016)04-0078-06

具有r个圈的仙人掌图关于距离-度指数的极值

王雨溪，陈翰麟，邓汉元*

(湖南师范大学数学与计算机科学学院，中国 长沙410081)

摘要设G=(V，E)是一个连通图.G的基于距离-度的拓扑指数一般定义为(u,v))，其中F=F(x,y,z)是一个函数,deg(u)是顶点u的度,d(u,v)是u和v之间的距离.若F分别是(x+y)z,xyz,(x+y)z-1和xyz-1，则IF(G)就分别是距离指数DD(G),Gutman指数Gut(G)，和加权Harary指数HA(G)与积加权Harary指数HM(G).本文确定了具有r个圈的仙人掌图关于和加权Harary指数与积加权Harary指数的最大值，以及关于度距离指数与Gutman指数的最小值；并刻画了对应的极图.

关键词仙人掌图；度距离指数；Gutman指数；和加权Harary指数；积加权Harary指数；极值