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Geometric Antipodal Distance-Regular Graphs with a Given Smallest Eigenvalue

  • Sejeong BangEmail author
Original Paper
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Abstract

In this paper we show that for a given integer \(m\ge 3\), there are only finitely many antipodal distance-regular graphs with \(c_2\ge 2\), odd diameter at least 5 and smallest eigenvalue at least \(-m\). To prove this we first show that for a given integer \(m\ge 3\), there are only finitely many antipodal distance-regular graphs with \(c_2\ge 2\), diameter at least 5 and smallest eigenvalue at least \(-m\) such that folded graphs are non-geometric. (A non-complete distance-regular graph is called geometric if it is the point graph of a partial linear space in which the set of lines is a set of Delsarte cliques.). Moreover, we show tight bounds for the diameter of geometric antipodal distance-regular graphs with an induced subgraph \(K_{2,1,1}\) by studying parameters for geometric antipodal distance-regular graphs.

Keywords

Distance-regular graph Geometric Antipodal Smallest eigenvalue Diameter bound Induced subgraph \(K_{2, 1, 1}\) 

Mathematics Subject Classification

05C12 05C50 05C62 05E30 

Notes

Acknowledgements

This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (NRF-2017R1D1A1B06029987). The author would like to thank the anonymous referees for their comments as their comments improved the paper.

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Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsYeungnam UniversityGyeongbukRepublic of Korea

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