Characterization of integral graphs

Establish a complete characterization of integral graphs, including restricted classes such as trees and unicyclic graphs, for which no complete characterization is currently known.

Background

The paper situates its results within the longstanding problem of characterizing graphs whose adjacency eigenvalues are all integers. Although substantial progress has been made, the authors state that a complete characterization remains unavailable even for the restricted families of trees and unicyclic graphs. The paper addresses a narrower structural question concerning odd cycles in integral unicyclic graphs, rather than resolving the broader characterization problem.