Distance-regularity and the spectrum of graphs Original Research Article

We deal with the question: Can one see from the spectrum of a graph Г whether it is distance-regular or not? Up till now the answer has not been known when Г has precisely four distinct eigenvalues (the diameter 3 case). We show that in this case the answer is negative. We also give positive answers in some special situations. For instance, if Г has the spectrum of a distance-regular graph with diameter 3 and μ = 1, then Г is distance-regular. Our main tools are eigenvalue techniques for partitioned matrices.