Characterization of graphs with eigenvalues bounded below

A core problem in spectral graph theory is the characterization of graphs with bounded adjacency eigenvalues. We define G(λ) to be the set of all graphs with smallest adjacency eigenvalue at least -λ. It is well-known that two infinite families of graphs namely line  graphs and generalized line graphs have adjacency values at least -2, i.e. they lie in  G(2). In 1976, Cameron, Goethals, Seidel, and Shult gave a complete characterization  of graphs in G(2). Recently, Jiang and Polyanskii proved that the number of connected  graphs in G(λ)\G(2) is finite iff λ ∈ (2, λ*) where λ* ≈ 2.0198008871. This tells us that  λ*  is the smallest λ > 2 such that G(λ)\G(2) contains infinitely many connected graphs. We will give a complete characterization of graphs in G(λ*)\G(2). In particular, we will  discuss the structure of large graphs in G(λ*)\G(2). This talk is based on joint work  with Dr. Zilin Jiang in arxiv 2404.13136.*