On Spectral Graph Determination

Abstract

The study of spectral graph determination is a fascinating area of research in spectral graph theory and algebraic combinatorics. This field focuses on examining the spectral characterization of various classes of graphs, developing methods to construct or distinguish cospectral nonisomorphic graphs, and analyzing the conditions under which a graph’s spectrum uniquely determines its structure. This paper presents an overview of both classical and recent advancements in these topics, along with newly obtained proofs of some existing results, which offer additional insights.

Keywords

• spectral graph theory
• spectral graph determination
• cospectral nonisomorphic graphs
• Haemers’ conjecture
• Turán graphs
• graph operations