The integral 3-harmonic graphs

Let G be a graph on n vertices v1, … , vn and let d(vi) be the degree of the vertex vi. If d(G) = (d(v1), … , d(vn))⊤ is an eigenvector of the (0, 1)-adjacency matrix A of G, i.e. A(G)d(G) = λd(G), then G is said to be λ-harmonic. In this paper all connected integral 3-harmonic graphs are determined. There are exactly 26 such graphs.