Let G be a graph on n vertices v 1 ,v 2 ,...,v n and let d(v i ) be the degree (= number of first neighbors) of the vertex v i . If (d(v 1 ),d(v 2 ),...,d(v n )) t is an eigenvector of the (0,1)-adjacency matrix of G, then G is said to be harmonic. Earlier all harmonic trees were determined; their number is infinite. We now show that for any c,c>1, the number of connected harmonic graphs with cyclomatic number c is finite. In particular, there are no connected non-regular unicyclic and bicyclic harmonic graphs and there exist exactly four and eighteen connected non-regular tricyclic and tetracyclic harmonic graphs.