關于圖的能量及擴展能量

周后卿

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關于圖的能量及擴展能量

周后卿

(邵陽學院 理學院，湖南 邵陽 422000)

圖G的能量是指圖G的鄰接矩陣特征值的絕對值之和﹒簡要介紹近幾年來國內外學者對能量以及擴展能量的研究情況和他們所取得的成果；重點介紹了幾類擴展能量，譬如預解能量、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；擴展能量

對于能量研究及應用，文獻[5-6]以及李學良、Yongtang Shi，Ivan Gutman合著的著作《Graph Energy》[7]集中體現了這方面的主要工作﹒在研究能量的基礎上，國內外學者如I. Gutman，K. C. Das，O. Rojo，B. Furtula，李學良、周波等將能量的概念推廣到所有簡單圖，定義了一系列的與圖的能量相類似的不變量，也即圖的擴展能量﹒

1 某些擴展能量的背景

2 一些擴展能量的界

關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內外學者研究的一個熱門話題﹒文獻[6]從研究者數量、研究人員分布、論文數量等指標統計了近20年來，關于能量研究的一些狀況，發現有63種能量已被研究，這里所介紹的只是其中很少的部分﹒對于能量、拉普拉斯能量以及無符號拉普拉斯能量，研究的人數最多、時間最長、成果最多，這里不多贅述﹒

下面介紹幾個擴展能量的研究情況﹒

2.1 預解能量(Resolvent energy)

文獻[29]給出了預解能量的一個上界和下界，證明了下列定理﹒

對于2部圖給出了一個上界，有下列結論﹒

文獻[30]討論了單圈圖、雙圈圖以及3圈圖的預解能量，證得了下面的一些結論﹒

研究者在文獻[30]中還討論了圖的預解能量的一些極值性質﹒

2.2 塞德爾能量(Seidel energy)

文獻[26]還就非共譜的等Seidel能量圖進行了分析，證明了下列定理﹒

同時，對于正則圖，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[40]研究了有向圖的斜能量的界，得到了下面這個結果﹒

文獻[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的話，那么它一定是一個正偶數；還推出了每一個正偶數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 結束語

本文著重介紹了一些擴展能量的研究成果，限于篇幅，還有許多能量沒有介紹﹒對能量的研究方法既有代數方法、矩陣論的方法；也有分析方法，利用不等式的技巧；圖論方法，對圖形做適當形變，限制圖的一些參數﹒借助計算機技術和軟件，從中發現規律，找出問題的解法﹒

總而言之，對圖的能量研究，文章雖然很多，但絕大多數傾向于圖的結構性質，而對它的應用研究得少﹒雖然也有部分學者在圖的能量應用方面做了一些研究，如文獻[46]探討了圖的能量在定量結構-性質/活性關系(QSPR/QSAR)中發揮的作用；文獻[47]說明圖的能量與熵有關；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揮作用[49]﹒但這些還遠遠不夠，因為能量在應用方面的研究結果少之又少﹒文獻[40]最后提了一個這樣的問題：是否能解釋斜能量在化學和其他學科中的應用？其實不止是斜能量，I. Gutman介紹了至今有63種能量被研究﹒那么，它們在生物、化學等學科中究竟有什么作用，這是一個值得深入探究的課題﹒

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(責任編校：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用研究﹒E-mail: zhouhq2004@163.com