For an integer \(p\ge 3\) , we study graphs whose adjacency spectral radius satisfies \(\rho (G)<\frac{p}{\sqrt{p-1}}=A_p\) or whose signless Laplacian spectral radius satisfies \(\kappa (G)<\frac{p^2}{p-1}=Q_p\) . 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 \(A_3<\rho (G)<A_4\) or \(Q_3<\kappa (G)<Q_4\) is much richer than the structure of graphs for which \(\rho (G)<A_3\) or \(\kappa (G)<Q_3\) , whose study has been initiated by Woo and Neumaier (Graphs Combin 23:713–726, 2007).
