Graphs with least eigenvalue −2; a historical survey and recent developments in maximal exceptional graphs Original Research Article

We survey the main results of the theory of graphs with least eigenvalue −2 starting from late 1950s (papers by A.J. Hoffman et al.), via important results (P.J. Cameron et al., J. Algebra 43 (1976) 305) involving root systems, to the recent approach by the star complement technique which culminated in finding and characterizing maximal exceptional graphs. Some novel results on maximal exceptional graphs are included as well. In particular, we show that all exceptional graphs, except for the cone over L(K8), can be obtained by the star complement technique starting from a unique (exceptional) star complement for the eigenvalue −2.