Graphs with few distinct eigenvalues and extremal energy

In his survey "Beyond graph energy: Norms of graphs and matrices" [23], 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 singular values 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 satisfyingthe properties required in the problems are integral, except for complete bipartite graphs K-p,K-q and disconnected graphs with a component K-p,K-q, where pq is not a perfect square. (C) 2021 Elsevier Inc. All rights reserved.