Geometric Antipodal Distance-Regular Graphs with a Given Smallest Eigenvalue

  • Sejeong BangEmail author
Original Paper


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.


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

Mathematics Subject Classification

05C12 05C50 05C62 05E30 



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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