Abstract
In 1986, Terwilliger showed that there is a strong relation between the eigenvalues of a distance-regular graph and the eigenvalues of a local graph. In particular, he showed that the eigenvalues of a local graph are bounded in terms of the eigenvalues of a distance-regular graph, and he also showed that if an eigenvalue \(\theta \) of the distance-regular graph has multiplicity m less than its valency k, then \(-1- \frac{b_1}{\theta +1}\) is an eigenvalue for any local graph with multiplicity at least \(k-m\). In this paper, we are going to generalize the results of Terwilliger to a broader class of subgraphs. Instead of local graphs, we consider induced subgraphs of distance-regular graphs (and more generally t-walk-regular graphs with t a positive integer) which satisfies certain regularity conditions. Then we apply the results to obtain bounds on eigenvalues of t-walk-regular graphs if the girth equals 3, 4, 5, 6 or 8. In particular, we will show that the second largest eigenvalue of a distance-regular graph with girth 6 and valency k is at most \(k-1\), and we will show that the only such graphs having \(k-1\) as its second largest eigenvalue are the doubled Odd graphs.
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Acknowledgments
S. Bang was supported by the 2014 Yeungnam University Research Grant. J. H. Koolen is partially supported by the ‘100 talents’ program of the Chinese Academy of Sciences and he is also partially supported by the National Natural Science Foundation of China (No. 11471009). J. Park was a postdoc at the School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, 230026, P. R. China, while the research for this paper was conducted. We also thank the anonymous referees for their comments. Their comments improved the presentation of the paper.
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Bang, S., Koolen, J.H. & Park, J. Some Results on the Eigenvalues of Distance-Regular Graphs. Graphs and Combinatorics 31, 1841–1853 (2015). https://doi.org/10.1007/s00373-015-1622-6
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DOI: https://doi.org/10.1007/s00373-015-1622-6