Abstract
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.
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