This is a survey paper on the second largest eigenvalue λ₂ of the adjacency matrix of a graph. Among the topics presented are the graphs with small λ₂, bounds for λ₂, algebraic connectivity, graphs ...

We investigate the rank of the adjacency matrix of large diluted random graphs: for a sequence of graphs (G n ) n≥0 converging locally to a Galton—Watson tree T (GWT), we provide an explicit formula ...