In this paper, we show that a connected graph with smallest eigenvalue at least −3 and large enough minimal degree is 2-integrable. This result generalizes a 1977 result of Hoffman for connected graphs with smallest eigenvalue at least −2.
In 1977, Hoffman gave a characterization of graphs with smallest eigenvalue at least −2. In this paper we generalize this result to graphs with smaller smallest eigenvalue. For the proof, we use a combinatorial object named Hoffman graph, introduced by Woo and Neumaier in 1995. Our result says that for every λ ≤ −2, if a graph with smallest eigenvalue at least λ satisfies some local conditions, then it is highly structured. We apply our result to graphs which are cospectral with the Hamming graph H(3, q), the Johnson graph J(v, 3) and the 2clique extension of grids, respectively.
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