Extremal graphs for the sum of two largest eigenvalues

Shaowei Sun; Yaping Min; Kinkar Chandra Das; Shaowei Sun; Yaping Min; Kinkar Chandra Das

doi:10.3934/math.2026618

AIMS Mathematics

2026, Volume 11, Issue 5: 15028-15036. doi: 10.3934/math.2026618

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Extremal graphs for the sum of two largest eigenvalues

1.
School of Science, Zhejiang University of Science and Technology, Hangzhou, Zhejiang 310023, China
2.
Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea

Received: 12 March 2026 Revised: 19 May 2026 Accepted: 26 May 2026 Published: 28 May 2026
MSC : 05C50, 15A18

In this paper, we characterize all connected graphs for which the sum of two largest eigenvalues is less than $ 4 $. As an application, we prove that the path graph minimizes this sum among all connected graphs of order $ n \geq 467 $, thereby solving a conjecture posed by Kumar, Liu, Monterde, Pragada, and Tait in "Maximum spectral sum of graphs (arXiv: 2604.00512v2)".
- eigenvalue sum,
- adjacency matrix,
- graph spectrum,
- path graph
Citation: Shaowei Sun, Yaping Min, Kinkar Chandra Das. Extremal graphs for the sum of two largest eigenvalues[J]. AIMS Mathematics, 2026, 11(5): 15028-15036. doi: 10.3934/math.2026618

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Abstract

In this paper, we characterize all connected graphs for which the sum of two largest eigenvalues is less than $ 4 $. As an application, we prove that the path graph minimizes this sum among all connected graphs of order $ n \geq 467 $, thereby solving a conjecture posed by Kumar, Liu, Monterde, Pragada, and Tait in "Maximum spectral sum of graphs (arXiv: 2604.00512v2)".

References

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