劉中柱

（惠州學(xué)院 數(shù)學(xué)與大數(shù)據(jù)學(xué)院,廣東 惠州 516007）

給定直徑圖的第1類Zagreb指數(shù)的上界及極圖

劉中柱

（惠州學(xué)院 數(shù)學(xué)與大數(shù)據(jù)學(xué)院,廣東 惠州 516007）

令圖G是具有n個(gè)頂點(diǎn)、直徑為d的簡(jiǎn)單連通圖.本文根據(jù)Ore的方法，給出了G的第1類Zagreb指數(shù)的上界，并刻畫(huà)了極圖.

Zagreb指數(shù)；簡(jiǎn)單圖；直徑

1 基本知識(shí)介紹

令G=(V,E)是簡(jiǎn)單連通圖，其中V,E分別是G的點(diǎn)集與邊集，且 ||V=n, ||E=m.圖G第1類和第2類Zagreb指數(shù)分別定義為

其中d(v)表示頂點(diǎn)v的度.

Zagreb指數(shù)是由Gutman和Trinajsti?[1]提出的經(jīng)典圖論拓?fù)渲笖?shù)，在分子設(shè)計(jì)、分子復(fù)雜性、能量等方面得到廣泛應(yīng)用，它反映了分子骨架的分支數(shù).Zagreb拓?fù)渲笖?shù)的數(shù)學(xué)性質(zhì)在文獻(xiàn)[2-3]中有相關(guān)研究.有關(guān)Zagreb指數(shù)的進(jìn)展及更多的研究背景可參考文獻(xiàn)[4-9]。鄧漢元[10]給出了樹(shù)、單圈圖與雙圈圖的Zagreb指數(shù)的結(jié)果；李樹(shù)超[11]給出了給定直徑的二部圖的第2類Zagreb指數(shù)的界并刻畫(huà)了極圖；最近關(guān)于Zagreb指數(shù)的結(jié)果參考文獻(xiàn)[12-16].

令G(n,d)為具有n個(gè)頂點(diǎn)，直徑為d≥2的簡(jiǎn)單連通圖.d(u,v)表示頂點(diǎn)u和v間的距離，G[E]表示邊集E在原圖G中的邊生成子圖，NG(v)表示點(diǎn)v在圖G中的鄰接點(diǎn)集，表示點(diǎn)集V1與V2間的邊的集合，E(V1,V2)表示端點(diǎn)分別在點(diǎn)集V1和V2間的邊的集合.令Kn是n個(gè)頂點(diǎn)的完全圖，Kn-e是在Kn中刪去一邊得到的圖，聯(lián)接1條懸掛邊到Kn-1-e中度為n-2的點(diǎn)上，得到的圖記為Kn,3,它是頂點(diǎn)數(shù)為n、直徑長(zhǎng)為3的圖.令P=v0v1…vd是長(zhǎng)為d的路，在頂點(diǎn)vi,vi+1,vi+2與Kn-d-1中所有的頂點(diǎn)連邊得到一系列圖，記為圖類Gn,d,其中1≤i≤d-3.

本文根據(jù)Ore[17]的方法與技巧，得到G(n,d)中第1類Zagreb指數(shù)的上界，并刻畫(huà)了極圖.

2 主要結(jié)論

引理1 令a≥a0≥a1≥…≥ad為非負(fù)整數(shù)，且當(dāng)且僅當(dāng)a0=a1=a2=a,且ai=0,3≤i≤d時(shí)等號(hào)成立.

從而存在另一個(gè)d+1元非負(fù)整數(shù)集π′={ } a0,…,ak-2,ak-1+1,ak-1,0,…,0,使得 f(π′)>f(π),與假設(shè)矛盾等式成立時(shí)當(dāng)且僅當(dāng)a0=a1=a2=a,且ai=0,3≤i≤d.

根據(jù)文獻(xiàn)[17]，可以得到定理1.

定理1 令G∈G(n,d),其中n≥4,從而有

1)當(dāng)d=2,M1(G)≤(n2-3)(n-2),當(dāng)且僅當(dāng)G=Kn-e時(shí)等號(hào)成立；

2)當(dāng)d=3,M1(G)=n3-5n2+6n+2,當(dāng)且僅當(dāng)G=Kn,3時(shí)等號(hào)成立；

證 令G∈G(n,d).令P=v0v1…vd為圖G的直徑路，Z=V-V(P).

當(dāng)d≥3時(shí)，2≤x0+xd≤n-d-1;

當(dāng)d=2時(shí)，有M1≤(n2-3)(n-2),當(dāng)且僅當(dāng)G=Kn-e,x0=xd=n-3時(shí)等號(hào)成立；

當(dāng)d=3時(shí)，有M1=n3-5n2+6n+2,當(dāng)且僅當(dāng)G=Kn,3,x0=n-4,xd=0或x0=0,xd=n-4時(shí)等號(hào)成立；

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[3] GUTMAN I,RUSCIC B,Trinajstiic N,et al.Graph theory and molecular orbitals XII Acyclic polyenes[J].J Chem Phys,1975,62: 3 399-3 405.

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The Upper Bound of the First Zagreb Index of Graphs with Given Diameter

LIU Zhongzhu

（Shool of Mathematics and Big Data Science,Huizhou University,Huizhou,Guangdong 516007,China）

LetGbe the graph withnvertices and diameterd.The idea of O.Orea(1986)is applied to investigate the upper bound of the first Zagreb index ofGand then characterize the extreme graph.

:Zagreb index；simple graph；diameter

O157.5

A

1009-8445（2017）02-0012-03

（責(zé)任編輯:陳 靜）

2016-12-19

惠州市科學(xué)技術(shù)創(chuàng)新基金資助項(xiàng)目(2014B020004027)；廣東省杰出青年教師基金資助項(xiàng)目(YQ2015155)；國(guó)家社會(huì)科學(xué)基金資助項(xiàng)目(15BTJ024）

劉中柱(1982-）,男,廣東河源人,惠州學(xué)院數(shù)學(xué)與大數(shù)據(jù)學(xué)院講師,博士.