關(guān)于圖的能量及擴(kuò)展能量

周后卿

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關(guān)于圖的能量及擴(kuò)展能量

周后卿

(邵陽(yáng)學(xué)院 理學(xué)院，湖南 邵陽(yáng) 422000)

圖G的能量是指圖G的鄰接矩陣特征值的絕對(duì)值之和﹒簡(jiǎn)要介紹近幾年來國(guó)內(nèi)外學(xué)者對(duì)能量以及擴(kuò)展能量的研究情況和他們所取得的成果；重點(diǎn)介紹了幾類擴(kuò)展能量，譬如預(yù)解能量、塞德爾能量、埃爾米特能量以及斜能量的研究成果；同時(shí)提出了在能量研究中存在的某些問題以及今后需要努力的一些方向﹒

特征值；能量；擴(kuò)展能量

對(duì)于能量研究及應(yīng)用，文獻(xiàn)[5-6]以及李學(xué)良、Yongtang Shi，Ivan Gutman合著的著作《Graph Energy》[7]集中體現(xiàn)了這方面的主要工作﹒在研究能量的基礎(chǔ)上，國(guó)內(nèi)外學(xué)者如I. Gutman，K. C. Das，O. Rojo，B. Furtula，李學(xué)良、周波等將能量的概念推廣到所有簡(jiǎn)單圖，定義了一系列的與圖的能量相類似的不變量，也即圖的擴(kuò)展能量﹒

1 某些擴(kuò)展能量的背景

2 一些擴(kuò)展能量的界

關(guān)于能量以及由此引申、類比、分化出來的其他能量，還有探索具有某極值能量的極圖是國(guó)內(nèi)外學(xué)者研究的一個(gè)熱門話題﹒文獻(xiàn)[6]從研究者數(shù)量、研究人員分布、論文數(shù)量等指標(biāo)統(tǒng)計(jì)了近20年來，關(guān)于能量研究的一些狀況，發(fā)現(xiàn)有63種能量已被研究，這里所介紹的只是其中很少的部分﹒對(duì)于能量、拉普拉斯能量以及無符號(hào)拉普拉斯能量，研究的人數(shù)最多、時(shí)間最長(zhǎng)、成果最多，這里不多贅述﹒

下面介紹幾個(gè)擴(kuò)展能量的研究情況﹒

2.1 預(yù)解能量(Resolvent energy)

文獻(xiàn)[29]給出了預(yù)解能量的一個(gè)上界和下界，證明了下列定理﹒

對(duì)于2部圖給出了一個(gè)上界，有下列結(jié)論﹒

文獻(xiàn)[30]討論了單圈圖、雙圈圖以及3圈圖的預(yù)解能量，證得了下面的一些結(jié)論﹒

研究者在文獻(xiàn)[30]中還討論了圖的預(yù)解能量的一些極值性質(zhì)﹒

2.2 塞德爾能量(Seidel energy)

文獻(xiàn)[26]還就非共譜的等Seidel能量圖進(jìn)行了分析，證明了下列定理﹒

同時(shí)，對(duì)于正則圖，證明了下面的定理﹒

2.3 埃爾米特能量(Hermitian energy)

2.4 斜能量(Skew energy)

文獻(xiàn)[40]研究了有向圖的斜能量的界，得到了下面這個(gè)結(jié)果﹒

文獻(xiàn)[40]還刻畫了具有最大斜能量的有向圖族，并且證明了有向圖的斜能量如果是有理數(shù)的話，那么它一定是一個(gè)正偶數(shù)；還推出了每一個(gè)正偶數(shù)一定是有向星圖的斜能量﹒文章最后提出了如下一些公開問題﹒

3 結(jié)束語(yǔ)

本文著重介紹了一些擴(kuò)展能量的研究成果，限于篇幅，還有許多能量沒有介紹﹒對(duì)能量的研究方法既有代數(shù)方法、矩陣論的方法；也有分析方法，利用不等式的技巧；圖論方法，對(duì)圖形做適當(dāng)形變，限制圖的一些參數(shù)﹒借助計(jì)算機(jī)技術(shù)和軟件，從中發(fā)現(xiàn)規(guī)律，找出問題的解法﹒

總而言之，對(duì)圖的能量研究，文章雖然很多，但絕大多數(shù)傾向于圖的結(jié)構(gòu)性質(zhì)，而對(duì)它的應(yīng)用研究得少﹒雖然也有部分學(xué)者在圖的能量應(yīng)用方面做了一些研究，如文獻(xiàn)[46]探討了圖的能量在定量結(jié)構(gòu)-性質(zhì)/活性關(guān)系(QSPR/QSAR)中發(fā)揮的作用；文獻(xiàn)[47]說明圖的能量與熵有關(guān)；能量在探尋阿爾茨海默病的遺傳原因[48]、流行病傳播模型研究中也發(fā)揮作用[49]﹒但這些還遠(yuǎn)遠(yuǎn)不夠，因?yàn)槟芰吭趹?yīng)用方面的研究結(jié)果少之又少﹒文獻(xiàn)[40]最后提了一個(gè)這樣的問題：是否能解釋斜能量在化學(xué)和其他學(xué)科中的應(yīng)用？其實(shí)不止是斜能量，I. Gutman介紹了至今有63種能量被研究﹒那么，它們?cè)谏?、化學(xué)等學(xué)科中究竟有什么作用，這是一個(gè)值得深入探究的課題﹒

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(責(zé)任編校：龔倫峰)

On Energy and Extended Energy of Graphs

ZHOU Houqing

(College of Science, Shaoyang University, Shaoyang, Hunan 422000, China)

The energy of a graph G is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. This paper introduces the research situation and achievements on energy and extended erengy of graphs at home and abroad in the past few years, it focuses on a few class extended energy, such as resolvent energy, Seidel energy, Hermitian energy and skew energy. At the same time the author also puts forward some existing problems, as well as pointing out some direction in the future.

eigenvalue; energy; extended energy

O157.5

A

10.3969/j.issn.1672-7304.2017.06.0009

1672–7304(2017)04–0040–06

2017-11-12

湖南省教育廳科研項(xiàng)目(15C1235)

周后卿(1963- )，男，湖南新邵人，教授，碩士，主要從事圖論及其應(yīng)用研究﹒E-mail: zhouhq2004@163.com