Finite forbidden subgraph characterization of graphs with bounded spectral radius

The study of eigenvalues associated with the adjacency matrix of graphs has intrigued mathematicians for decades and has given rise to the dynamic field of Spectral Graph Theory. Within this field, a particular set of problems has piqued the interest of graph theorists: characterizing graphs with a bounded spectral radius. We define F(λ) to be the set of all graphs with spectral radius at most λ, then one can completely describe F(λ) by providing a set of forbidden subgraphs, known as forbidden subgraph characterization of the set. In this talk, we will discuss the recent results by Jiang and Polyanskii, who provide sufficient and necessary conditions for the set F(λ) to have a finite forbidden subgraph characterization.