Graphs whose spectral radius is bounded by a fixed Hoffman–Smith limit point - Borba, Elizandro Max; Calegari, Rafael; Hoppen, Carlos; Trevisan, Vilmar; Veloso, Bruno Scaratti (Journal of algebraic combinatorics, 2025)
For an integer p ≥ 3 , we study graphs whose adjacency spectral radius satisfies ρ ( G ) < p p - 1 = A p or whose signless Laplacian spectral radius satisfies κ ( G ) < p 2 p - 1 = Q p . The numbers A p and Q p are known as Hoffman–Smith limit points. For general p , we find upper bounds on the maximum degree of such graphs G and describe the graphs for which the upper bound is achieved with respect to the adjacency matrix. Moreover, we describe forbidden substructures for graphs in these classes. For p = 4 , we show that the structure of graphs G such that A 3 < ρ ( G ) < A 4 or Q 3 < κ ( G ) < Q 4 is much richer than the structure of graphs for which ρ ( G ) < A 3 or κ ( G ) < Q 3 , whose study has been initiated by Woo and Neumaier (Graphs Combin 23:713–726, 2007).