Abstract
Let \({\varGamma }\) be a non-bipartite distance-regular graphs with valancy k, diameter D and a smallest eigenvalue \(\theta _{\mathrm{min}}\). In 2019, Qiao, Jing and Koolen classified the non-bipartite distance-regular graphs with \(\theta _{\mathrm{min}} \le -\frac{D-1}{D}k\) for \(D=4, 5\). In this paper, we classify the non-bipartite distance-regular graphs with \(\theta _{\mathrm{min}}\le -\frac{D-2}{D-1}k\) for \(D=5, 6\). We remark that the technique of this paper is an extension of the approach taken by Qiao, Jing and Koolen on the study of non-bipartite distance-regular graphs in 2019.
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References
Abdollahi, A., van Dam, E.R., Jazaeri, M.: Distance-regular Cayley graphs with least eigenvalue \(-2\). Des. Codes Cryptogr. 84, 73–85 (2017)
Bang, S.: Distance-regular graphs with an eigenvalue \(-k<\theta \le 2-k\). Electron. J. Combin. 21, P1.4 (2014)
Bang, S., Dubickas, A., Koolen, J.H., Moulton, V.: There are only finitely many distance-regular graphs of fixed valency greater than two. Adv. Math. 269, 1–55 (2015)
Bang, S., Koolen, J.H., Park, J.: Some results on the eigenvalues of distance-regular graphs. Graphs Combin. 31, 1841–1853 (2015)
Biggs, N.L., Boshier, A.G., Shawe-Taylor, J.: Cubic distance-regular graphs. J. London Math. Soc. 2(33), 385–394 (1986)
Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-Regular Graphs. Springer, Berlin (1989)
Brouwer, A.E., Haemers, W.H.: Spectra of Graphs. Springer, New York (2012)
Brouwer, A.E., Koolen, J.H.: The distance-regular graphs of valency four. J. Algebraic Combin. 10, 5–24 (1999)
van Dam, E.R., Koolen, J.H., Tanaka, H.: Distance-regular graphs. Electron. J. Combin. \(\#\)DS22 (2016)
Knox, F., Mohar, B.: Fractional decompositions and the smallest-eigenvalue separation. Electron. J. Combin. 26(4), \(\#\)P4.41 (2019)
Koolen, J.H., Yu, H.: The distance-regular graphs such that all of its second largest local eigenvalues are at most one. Linear Algebra Appl. 435, 2507–2519 (2011)
Koolen, J.H., Park, J., Yu, H.: An inequality involving the second largest and smallest eigenvalue of a distance-regular graph. Linear Algebra Appl. 434, 2404–2412 (2011)
Qiao, Z., Koolen, J.H.: A valency bound for distance-regular graphs. J. Combin. Theory Ser. A 155, 304–320 (2018)
Qiao, Z., Jing, Y.F., Koolen, J.H.: Non-bipartite distance-regular graphs with a small smallest eigenvalue. Electron. J. Combin. 26(2) \(\#\)P2.41 (2019)
Seidel, J.J.: Strongly regular graphs with \((-1, 1, 0)\) adjacency matrix having eigenvalue \(3\). Linear Algebra Appl. 1, 281–298 (1968)
Acknowledgements
The authors would like to thank the referees for giving this paper a careful reading and many valuable comments and useful suggestions. This work was supported by the NSF of China (No. 11971146, No. 12071344), the NSF of Hebei Province (No. A2019205089).
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Li, J., Wang, Y., Hou, B. et al. Non-Bipartite Distance-Regular Graphs with Diameters 5, 6 and a Smallest Eigenvalue. Graphs and Combinatorics 38, 55 (2022). https://doi.org/10.1007/s00373-022-02458-2
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DOI: https://doi.org/10.1007/s00373-022-02458-2