On acyclic conjugated molecules with minimal energies

The energy of a graph is defined as the sum of the absolute values of all e igenvalues of the graph. In [5] Gutman (J. Math. Chem.1 (1987) 123-143) pro poses two conjectures about the minimum of the energy of conjugated trees ( trees with a perfect matching). This paper mathematically verifies the two conjectures. In addition, trees with the second and the third smallest ener gies are also discussed. (C) 1999 Published by Elsevier Science B.V. All ri ghts reserved.