Developments on the Hoffman program of graphs

For each squared graph matrix M, the Hoffman program consists of two aspects: finding all the possible limit points of M-spectral radii of graphs and detecting all the connected graphs whose M-spectral radius does not exceed a fixed limit point. In this survey, we summarize the results on this topic concerning several graph matrices, including the adjacency, the Laplacian, the signless Laplacian, the Hermitian adjacency and the skew-adjacency matrix of graphs. The correspondent problems related to tensors of hypergraphs are also discussed. Moreover, we obtain new results about the Hoffman program with relation to the $A_{\alpha }$-matrix. In particular, we get two generalized versions of it applicable to nonnegative symmetric matrices with fractional elements. We also retrieve the limit points of spectral radii of (signless) Laplacian matrices of graphs less than $2+\frac{1}{3}({(54-6\sqrt{33})}^{\frac{1}{3}}+{(54+6\sqrt{33})}^{\frac{1}{3}})$.