Spectrum of Johnson graphs

In this paper we prove new bounds on the second eigenvalue of Johnson graphs. We then apply these bounds to obtain new results on the modularity of Johnson graphs and their random subgraphs, hamiltonicity of Johnson graphs and thresholds of the appearance of the hamilton cycles and giant components. Futhermore, we provide general bounds on vertex connectivity and the stability of the modularity for $(n,d,\lambda )$-graphs.