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 \(-2\).
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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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Huang, X., Huang, Q. & Lu, L. Graphs with at Most Three Distance Eigenvalues Different from \(-1\) and \(-2\). Graphs and Combinatorics 34, 395–414 (2018). https://doi.org/10.1007/s00373-018-1880-1
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DOI: https://doi.org/10.1007/s00373-018-1880-1