Extrema of graph eigenvalues

Abstract In 1993 Hong asked what are the best bounds on the k\u27th largest eigenvalue λk(G) of a graph G of order n. This challenging question has never been tackled for any 22, and even tighter bounds are obtained for the k\u27th largest singular value λ∗k(G). Some of these bounds are based on Taylor\u27s 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 λkappa;(G)+ λk(G¯) be? These constructions are successful also in another open question: How large can the Ky Fan norm λ∗1(G) +···+ λ∗k (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)-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