Graphs with at most two nonzero distinct absolute eigenvalues

TL;DR

This paper characterizes graphs with at most two nonzero absolute eigenvalues, providing solutions to open problems on graph energy bounds and identifying their spectral properties and specific families.

Contribution

It solves open problems by characterizing graphs with limited eigenvalue spectra related to energy bounds and identifies their structural and spectral properties.

Findings

Graphs have at most two nonzero absolute eigenvalues.

Most such graphs are characterized; infinite families are identified.

Graphs satisfying the properties are mostly integral, with specific exceptions.

Abstract

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.