Spectral characterizations of two families of nearly complete bipartite   graphs
  It is not hard to find many complete bipartite graphs which are not determined by their spectra. We show that the graph obtained by deleting an edge from a complete bipartite graph is determined by its spectrum. We provide some graphs, each of which is obtained from a complete bipartite graph by adding a vertex and an edge incident on the new vertex and an original vertex, which are not determined by their spectra. 
Chia-an Liu, Chih-wen Weng
math.CO