On the Integrability of Strongly Regular Graphs
- 121 Downloads
Koolen et al. showed that if a connected graph with smallest eigenvalue at least \(-3\) has large minimal valency, then it is 2-integrable. In this paper, we will prove that a lower bound for the minimal valency is 166.
KeywordsStrongly regular graph Lattice s-Integrability
Mathematics Subject Classification05C50 05E30 11H99
The first author J.H. Koolen is partially supported by the National Natural Science Foundation of China (Grant No. 11471009 and Grant No. 11671376) and Anhui Initiative in Quantum Information Technologies (No. AHY150000). The second author M.U. Rehman is supported by the Chinese Scholarship Council at USTC, China. The corresponding author Q. Yang is partially supported by the Chinese Scholarship Council (No. 201806340049). We also would like to thank Akihiro Munemasa for pointing out that one can extend the complement of the McLaughlin graph to obtain a slightly better lower bound.
- 3.Brouwer, A.E., van Lint, J.H.: Strongly regular graphs and partial geometries. Enumer. Des. 85, 122 (1982)Google Scholar
- 10.Deza, M., Grishukhin, V.P., Laurent, M.: Hypermetrics in Geometry of Numbers (DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 20), pp. 1–109. American Mathematical Socirty, Providence (1995)Google Scholar
- 21.Neumaier, A.: Strongly regular graphs with smallest eigenvalue \(-m\). Arch. Math. (Basel), 33 392–400 (1979/80)Google Scholar
- 24.Payne, S.E., Thas, J.A.: Finite generalized quadrangles. In: EMS Series of Lectures in Mathematics, second ed. European Mathematical Society (EMS), Zürich (2009)Google Scholar