Graphs and Combinatorics

, Volume 34, Issue 3, pp 395–414

# Graphs with at Most Three Distance Eigenvalues Different from $$-1$$ and $$-2$$

Article

## Abstract

Let G be a connected graph on n vertices, and let D(G) be the distance matrix of G. Let $$\partial _1(G)\ge \partial _2(G)\ge \cdots \ge \partial _n(G)$$ denote the eigenvalues of D(G). In this paper, we characterize all connected graphs with $$\partial _{3}(G)\le -1$$ and $$\partial _{n-1}(G)\ge -2$$. In the course of this investigation, we determine all connected graphs with at most three distance eigenvalues different from $$-1$$ and $$-2$$.

## Keywords

Distance matrix The third largest distance eigenvalue The second least distance eigenvalue

05C50

## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (11671344, 11701492). We are extremely grateful to the anonymous referees for their constructive comments and suggestions, which helped us to improve the manuscript.

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© Springer Japan KK, part of Springer Nature 2018

## Authors and Affiliations

• Xueyi Huang
• 1
• Qiongxiang Huang
• 1
• Lu Lu
• 1
1. 1.College of Mathematics and Systems ScienceXinjiang UniversityÜrümqiPeople’s Republic of China