Extrema of graph eigenvalues

In 1993 Hong asked what are the best bounds on the k  'th largest eigenvalue λk(G)λk(G) of a graph G of order n  . This challenging question has never been tackled for any 22k>2, and even tighter bounds are obtained for the k  'th largest singular value λk⁎(G).Some of these bounds are based on Taylor's strongly regular graphs, and others on a method of Kharaghani for constructing Hadamard matrices. The same kind of constructions are applied to other open problems, like Nordhaus–Gaddum problems of the kind: How large can  λk(G)+λk(G¯)be?These constructions are successful also in another open question: How large can the Ky Fan norm  λ1⁎(G)+⋯+λk⁎(G)be? Ky Fan norms of graphs generalize the concept of graph energy, so this question generalizes the problem for maximum energy graphs.In the final section, several results and problems are restated for (−1,1)(−1,1)-matrices, which seem to provide a more natural ground for such research than graphs.Many of the results in the paper are paired with open questions and problems for further study.