2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/142675We prove that if the spectral radius of a graph G of order n is larger than the spectral radius of the r-partite Turan graph of the same order, then G contains various supergraphs of the complete graph of order r+1. In particular G contains a complete r-partite graph of size log n with one edge added to the first part. These results complete a project of Erdos from 1963. We also give corresponding stability results.Combinatorics05C50,05C35text