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基于隔离集的不含给定特征值的图构造
| 更新时间：2026-07-08
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基于隔离集的不含给定特征值的图构造
Graph construction without a given eigenvalue based on isolation sets
- 中山大学学报(自然科学版)(中英文) 2026年 页码：1-7
- 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：
杨慧如（2002年生），女；研究方向：图谱理论； E-mail： 107622024210484@stu.xjnu.edu.cn
段芳（1979年生），女；研究方向：图谱理论； E-mail： 107622007010014@xjnu.edu.cn
- 基金信息：国家自然科学基金(12561064)
DOI：10.11714/acta.snus.ZR20260018
中图分类号： O157.5收稿：2026-01-14，
修回：2026-05-27，
录用：2026-05-28，
网络首发：2026-07-08，
- 稿件说明：
移动端阅览
杨慧如, 段芳, 李晓娟. 基于隔离集的不含给定特征值的图构造[J/OL]. 中山大学学报(自然科学版)(中英文), 2026,1-7.
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.
杨慧如, 段芳, 李晓娟. 基于隔离集的不含给定特征值的图构造[J/OL]. 中山大学学报(自然科学版)(中英文), 2026,1-7. DOI： 10.11714/acta.snus.ZR20260018.
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.
基于隔离集的不含给定特征值的图构造
摘要
设
<math id="M1"><mi>G</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418895&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418894&type=
2.03200006
2.28600001
是
<math id="M2"><mi>n</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418857&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418880&type=
1.60866666
2.28600001
阶简单图，
<math id="M3"><mi>μ</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418858&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418881&type=
2.03200006
2.87866688
是
<math id="M4"><mi>G</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418895&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418894&type=
2.03200006
2.28600001
的一个特征值， 其重数为
<math id="M5"><mi>k</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418874&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418882&type=
1.43933344
2.28600001
，
<math id="M6"><mi>H</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418875&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418883&type=
2.28600001
2.28600001
是删除
<math id="M7"><mi>G</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418895&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418894&type=
2.03200006
2.28600001
的
<math id="M8"><mi>k</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418874&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418882&type=
1.43933344
2.28600001
个点的集合
<math id="M9"><mi>X</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418861&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418884&type=
2.11666679
2.28600001
得到的
<math id="M10"><mi>G</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418895&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418894&type=
2.03200006
2.28600001
的导出子图. 若
<math id="M11"><mi>μ</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418862&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418899&type=
2.03200006
2.87866688
不是
<math id="M12"><mi>H</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418923&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418900&type=
2.28600001
2.28600001
的特征值， 那么就称
<math id="M13"><mi>H</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418923&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418900&type=
2.28600001
2.28600001
是图
<math id="M14"><mi>G</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418895&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418894&type=
2.03200006
2.28600001
关于特征值
<math id="M15"><mi>μ</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418862&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418899&type=
2.03200006
2.87866688
的星补，
<math id="M16"><mi>X</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418909&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418901&type=
2.11666679
2.28600001
是图
<math id="M17"><mi>G</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418895&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418894&type=
2.03200006
2.28600001
关于特征值
<math id="M18"><mi>μ</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418862&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418899&type=
2.03200006
2.87866688
的星集. 星集常用于研究某个给定特征值
<math id="M19"><mi>μ</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418862&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418899&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
<math id="M20"><mi>μ</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418862&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418899&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
<math id="M21"><mi>G</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418910&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418902&type=
2.37066650
2.62466669
be a simple graph，
<math id="M22"><mi>μ</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418903&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418925&type=
2.37066650
3.38666677
be an eigenvalue of
G
with multiplicity
k
， and
H
be the induced subgraph of
G
obtained by deleting a set
X
of
k
vertices. If
<math id="M23"><mi>μ</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418903&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418925&type=
2.37066650
3.38666677
is not an eigenvalue of
H
， then
H
is called the star complement in
G
for
<math id="M24"><mi>μ</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418903&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418925&type=
2.37066650
3.38666677
， and
X
is called the star set in
G
for
<math id="M25"><mi>μ</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418903&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418925&type=
2.37066650
3.38666677
. Star sets are commonly used to study the multiplicity problem for a given eigenvalue
<math id="M26"><mi>μ</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418903&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418925&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
<math id="M27"><mi>μ</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418903&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112418925&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 .
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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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