The Hoffman program of graphs: old and new

The Hoffman program with respect to any real or complex square matrix $M$ associated to a graph $G$ stems from A. J. Hoffman's pioneering work on the limit points for the spectral radius of adjacency matrices of graphs less than $\sqrt{2+\sqrt{5}}$. The 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 paper, we summarize the results on this topic concerning several graph matrices, including the adjacency, the Laplacian, the signless Laplacian, the Hermitian adjacency and skew-adjacency matrix of graphs. As well, the tensors of hypergraphs are discussed. Moreover, we obtain new results about the Hoffman program with relation to the $A_\alpha$-matrix. Some further problems on this topic are also proposed.