  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.
