Eigenvalue problems of Nordhaus–Gaddum type

Let G be a graph with n vertices and m   edges and let μ1(G)⩾⋯⩾μn(G)μ1(G)⩾⋯⩾μn(G) be the eigenvalues of its adjacency matrix. We discuss the following general problem. For k fixed and n large, find or estimatefk(n)=maxv(G)=n|μk(G)|+|μk(G¯)|.In particular, we prove that43n-2⩽f1(n)<(2-c)nfor some c>10-7c>10-7 independent of nn. We also show that 22n-3