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《山东大学学报(理学版)》 ›› 2023, Vol. 58 ›› Issue (11): 160-164.doi: 10.6040/j.issn.1671-9352.0.2022.011

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负惯性指数为2的图的结构刻画

王爽(),段芳*()   

  1. 新疆师范大学数学科学学院, 新疆 乌鲁木齐 830017
  • 收稿日期:2021-12-28 出版日期:2023-11-20 发布日期:2023-11-07
  • 通讯作者: 段芳 E-mail:61586369@qq.com;fangbing327@126.com
  • 作者简介:王爽(1978—),女,讲师,硕士,研究方向为偏微分方程数值解. E-mail:61586369@qq.com
  • 基金资助:
    新疆维吾尔自治区自然科学基金资助项目(2022D01A218);新疆师范大学基金资助项目(SDJG2021-01)

Complete characterization of graphs with exactly two negative eigenvalues

Shuang WANG(),Fang DUAN*()   

  1. School of Mathematics Science, Xinjiang Normal University, Urumqi 830017, Xinjiang, China
  • Received:2021-12-28 Online:2023-11-20 Published:2023-11-07
  • Contact: Fang DUAN E-mail:61586369@qq.com;fangbing327@126.com

摘要:

用符号Γn表示点数为n、负惯性指数为2的所有图构成的图集。利用定义的图的3种合同变换, 在图集Γn中可以找出8个零度不超过2的特殊的图, 使得图集Γn中其它图都可以由这8个特殊的图构造出来。因此, 负惯性指数为2的图得到了完整的刻画。

关键词: 负惯性指数, 正惯性指数, 零度, 合同变换

Abstract:

Let Γn be the set of graphs of order n such that each of them has exactly two negative eigenvalues. In this paper, we introduce 3 graph transformations. Furthermore, we find 8 specific graphs in Γn, each of them has nullity no more than two and we show that, in terms of the above three congruent transformations of graph, all the other graphs of Γn can be constructed from these 8 specific graphs. Hence we completely characterize the graphs with exactly two negative eigenvalues.

Key words: negative inertia index, positive inertia index, nullity, congruent transformation

中图分类号: 

  • O157.5

图1

图C6和H1-H9"

图2

图P4、P5、C5和G1-G3"

图3

图H10-H15"

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