Exactly thirteen connected cubic graphs have integral spectra

The problem of identifying those graphs whose spectra consist entirely of integers was first posed by F. Harary. We examined some elementary procedures for constructing integral graphs in [6]. Although the general problem seems intractible, it is easy to find the seven connected graphs with integral spectra, maximum degree at most three, and minimum degree less than three. This article was inspired by Cvetkovic's attempt [4] to find the connected cubic integral graphs. He had displayed twelve such graphs, and had restricted the remaining possibilities to ninety-five potential spectral.