Graph construction without a given eigenvalue based on isolation sets

Yang Huiru; Duan Fang; Li Xiaojuan

doi:10.11714/acta.snus.ZR20260018

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Graph construction without a given eigenvalue based on isolation sets

更新时间：2026-07-13

- Graph construction without a given eigenvalue based on isolation sets
- Acta Scientiarum Naturalium Universitatis Sunyatseni Pages: 1-7(2026)
- 作者机构：
  
  新疆师范大学数学科学学院，新疆乌鲁木齐 830017
- 作者简介：
- 基金信息：
- DOI：10.11714/acta.snus.ZR20260018
  CLC： O157.5
- Received：14 January 2026，
  
  Revised：2026-05-27，
  
  Accepted：28 May 2026，
  
  Online First：13 July 2026，
- 稿件说明：
移动端阅览
Yang Huiru, Duan Fang, Li Xiaojuan. Graph construction without a given eigenvalue based on isolation sets[J/OL]. Acta Scientiarum Naturalium Universitatis Sunyatseni, 2026, 1-7.
DOI：

Yang Huiru, Duan Fang, Li Xiaojuan. Graph construction without a given eigenvalue based on isolation sets[J/OL]. Acta Scientiarum Naturalium Universitatis Sunyatseni, 2026, 1-7. DOI： 10.11714/acta.snus.ZR20260018.

摘要

设

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821400&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821434&type=

2.03200006

2.28600001

是

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821422&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821414&type=

1.60866666

2.28600001

阶简单图，

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821401&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821435&type=

2.03200006

2.87866688

是

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821400&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821434&type=

2.03200006

2.28600001

的一个特征值，其重数为

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821423&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821415&type=

1.43933344

2.28600001

，

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821402&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821436&type=

2.28600001

是删除

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821400&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821434&type=

2.03200006

2.28600001

的

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821423&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821415&type=

1.43933344

2.28600001

个点的集合

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821424&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821416&type=

2.11666679

2.28600001

得到的

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821400&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821434&type=

2.03200006

2.28600001

的导出子图. 若

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821448&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821437&type=

2.03200006

2.87866688

不是

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821425&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821417&type=

2.28600001

的特征值，那么就称

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821425&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821417&type=

2.28600001

是图

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821400&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821434&type=

2.03200006

2.28600001

关于特征值

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821448&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821437&type=

2.03200006

2.87866688

的星补，

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821449&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821438&type=

2.11666679

2.28600001

是图

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821400&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821434&type=

2.03200006

2.28600001

关于特征值

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821448&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821437&type=

2.03200006

2.87866688

的星集. 星集常用于研究某个给定特征值

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821448&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821437&type=

2.03200006

2.87866688

的重数问题. 本文基于星集定义了两个特殊的隔离集，并讨论其在不以给定实数

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821448&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821437&type=

2.03200006

2.87866688

作为特征值的图的结构刻画问题中的作用.

Abstract

Let

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821426&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821463&type=

2.37066650

2.62466669

be a simple graph，

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821450&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821439&type=

2.37066650

3.38666677

be an eigenvalue of

with multiplicity

， and

be the induced subgraph of

obtained by deleting a set

vertices. If

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821450&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821439&type=

2.37066650

3.38666677

is not an eigenvalue of

， then

is called the star complement in

for

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821450&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821439&type=

2.37066650

3.38666677

， and

is called the star set in

for

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821450&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821439&type=

2.37066650

3.38666677

. Star sets are commonly used to study the multiplicity problem for a given eigenvalue

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821450&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821439&type=

2.37066650

3.38666677

. This paper defines two special isolation sets based on star sets and discusses their role in characterizing the structure of graphs

that do not have a given real number

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821450&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112821439&type=

2.37066650

3.38666677

as an eigenvalue.

关键词

Keywords

references

居余马， 2002 . 线性代数［M］. 2版 . 北京：清华大学出版社 .

Cvetković D ， Rowlinson P ， Simić S ， 1999 . Some characterizations of graphs by star complements ［J］. Linear Algebra Appl ， 301 （ 1/2/3 ）： 81 - 97 .

Cvetković D ， Rowlinson P ， Simić S ， 2004 . Spectral generalizations of line graphs： On graphs with least eigenvalue -2 ［M］. Cambridge ： Cambridge University Press .

Cvetković D ， Rowlinson P ， Simić S ， 2009 . An introduction to the theory of graph spectra ［M］. Cambridge ： Cambridge University Press .

Ellingham M N ， 1993 . Basic subgraphs and graph spectra ［J］. Australas J Comb ， 8 ： 247 - 266 .

Jackson P S ， Rowlinson P ， 1999 . On graphs with complete bipartite star complements ［J］. Linear Algebra Appl ， 298 （ 1/2/3 ）： 9 - 20 .

Rowlinson P ， 1993 . Eutactic stars and graph spectra ［M］//Brualdi R A， et al. Combinatorial and graph-theoretical problems in linear algebra . New York ： Springer .

Rowlinson P ， 2018 . An extension of the star complement technique for regular graphs ［J］. Linear Algebra Appl ， 557 ： 496 - 507 .

Rowlinson P ， Tayfeh-Rezaie B ， 2010 . Star complements in regular graphs： Old and new results ［J］. Linear Algebra Appl ， 432 （ 9 ）： 2230 - 2242 .

Stanić Z ， 2008 . Some star complements for the second largest eigenvalue of a graph ［J］. Ars Math Contemp ， 1 （ 2 ）： 126 - 136 .

Stanić Z ， 2025 . Employing star complements in search for graphs with fixed rank ［J］. Linear Algebra Appl ， 718 ： 14 - 29 .

Wang J ， Yuan X Y ， Liu L L ， 2019 . Regular graphs with a prescribed complete multipartite graph as a star complement ［J］. Linear Algebra Appl ， 579 ： 302 - 319 .

Yuan X Y ， Chen H Y ， Liu L L ， 2017 . On the characterization of graphs by star complements ［J］. Linear Algebra Appl ， 533 ： 491 - 506 .

Yuan X Y ， Zhao Q Q ， Liu L L ， et al . 2018 . On graphs with prescribed star complements ［J］. Linear Algebra Appl ， 559 ： 80 - 94 .

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