  In his survey "Beyond graph energy: Norms of graphs and matrices" (2016),
Nikiforov proposed two problems concerning characterizing the graphs that
attain equality in a lower bound and in a upper bound for the energy of a
graph, respectively. We show that these graphs have at most two nonzero
distinct absolute eigenvalues and investigate the proposed problems organizing
our study according to the type of spectrum they can have. In most cases all
graphs are characterized. Infinite families of graphs are given otherwise. We
also show that all graphs satifying the properties required in the problems are
integral, except for complete bipartite graphs $K_{p,q}$ and disconnected
graphs with a connected component $K_{p,q}$, where $pq$ is not a perfect
square.
