The Hoffman program of graphs: old and new

Abstract

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 2 + √ 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 Aα-matrix. Some further problems on this topic are also proposed.

A Hoffman's Theorem: a revisit with new discovery

“…Apart from Lemma 3.1, which is due to Nikiforov, all the other results given in this section without a proof are taken from the reference [19] written by the same authors of this paper.…”

“…Lemma 3.2. [19] Let uv be an edge of the connected graph G, and let G uv be the graph obtained from G by subdividing the edge uv of G. Set α ∈ [0, 1).…”

“…Lemma 3.4. [19] Let X and Y be two vertex-disjoint connected graphs, and let G n = XY (x, y; n) be the graph obtained by joining x ∈ V (X) and y ∈ V (Y ) by a path of length n + 1 (see Fig. 4).…”

“…Lemma 3.7. [19] The A α -spectral radius of the graph sequence {G u (P n )} n∈N has a limit point χ u (G) 2. Moreover,…”

“…The above investigation has been known as the Hoffman program of graphs with respect to the adjacency matrix. For more details on the Hoffman program of graphs, see [17,Section 1.3.3], [1,Section 3.3] and a new survey [19].…”