JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2023, Vol. 58 ›› Issue (11): 160-164.doi: 10.6040/j.issn.1671-9352.0.2022.011

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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

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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

CLC Number: 

  • O157.5

Fig.1

Graphs C6 and H1-H9"

Fig.2

Graphs P4, P5, C5 and G1-G3"

Fig.3

Graphs H10-H15"

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