Harmonic graphs with small number of cycles

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. (C) 2002 Elsevier Science B.V. All rights reserved.