Abstract The eigenvalues of a graph are the eigenvalues of its adjacency matrix. In this paper regular connected graphs with four eigenvalues, the least of which is — 2, are examined. Many line graphs are of this type and it is shown that, with only a finite number of exceptions, all graphs of this type are line graphs of strongly regular graphs. symmetric balanced incomplete block designs, or complete bipartite graphs. Certain of these graphs are characterized by their spectra among all graphs with the same number of vertices. We show that if the number of vertices is large enough, the existence of a graph with the same spectrum and number of vertices as the line graph of a complete bipartite graph and which is not isomorphic to it is equivalent to the existence of a symmetric Hadamard matrix with constant diagonal. Finally, we give some necessary and sufficient conditions for the line graph of a complete bipartite graph to be characterized by its spectrum and number of vertices.
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