# Geometric Antipodal Distance-Regular Graphs with a Given Smallest Eigenvalue

- 2 Downloads

## 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.

## References

- 1.Bang, S.: Diameter bounds for geometric distance-regular graphs. Discrete Math.
**341**, 253–260 (2018)MathSciNetCrossRefzbMATHGoogle Scholar - 2.Bang, S.: Geometric distance-regular graphs without \(4\)-claws. Linear Algebra Appl.
**438**, 37–46 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 3.Bang, S., Hiraki, A., Koolen, J.H.: Delsarte clique graphs. Eur. J. Combin.
**28**, 501–516 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Bang, S., Koolen, J.H.: A sufficient condition for distance-regular graphs to be geometric (in preparation, 2018)Google Scholar
- 5.Brouwer, A.E.: Distance regular graphs of diameter \(3\) and strongly regular graphs. Discrete Math.
**49**, 101–103 (1984)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-Regular Graphs. Springer, Berlin (1989)CrossRefzbMATHGoogle Scholar
- 7.Cameron, P.J., Goethals, J.-M., Seidel, J.J., Shult, E.E.: Line graphs, root systems, and elliptic geometry. J. Algebra
**43**, 305–327 (1976)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Delsarte, P.: An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl.
**10**, 1973 (1973)MathSciNetzbMATHGoogle Scholar - 9.Godsil, C.D.: Algebraic Combinatorics. Chapman and Hall, New York (1993)zbMATHGoogle Scholar
- 10.Godsil, C.D.: Geometric distance-regular covers. N. Z. J. Math.
**22**, 31–38 (1993)MathSciNetzbMATHGoogle Scholar - 11.Koolen, J.H., Bang, S.: On distance-regular graphs with smallest eigenvalue at least \(-m\). J. Combin. Theory Ser. B
**100**, 573–584 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Neumaier, A.: Strongly regular graphs with smallest eigenvalue \(m\). Arch. Math. (Basel)
**33**, 392–400 (1979). (1979-80)MathSciNetCrossRefzbMATHGoogle Scholar - 13.van Dam, E.R., Koolen, J.H., Tanaka, H.: Distance-regular graphs. Electron. J. Combin.
**DS22**, 1–156 (2016)zbMATHGoogle Scholar