SOME METHODS FOR COUNTING THE SPANNING-TREES IN LABELED MOLECULAR GRAPHS, EXAMINED IN RELATION TO CERTAIN FULLERENES

The number of spanning trees in a molecular graph (its 'complexity') h as been of recent interest and, in this paper, various methods are app lied to calculate the complexities of graphs that represent the fuller enes - as exemplified by the molecules C-60 and C-70, and the notional structures C-60 (known as 'handballene') and C-120 ('Archimedene'). T hese graphs are large, regular and highly symmetrical and the methods chosen address the computational difficulties and advantages presented by these features. The methods discussed are of general applicability when the graph under study has at least one of these properties. One of the methods needs only 'pencil-and-paper' working when applied to t he two C-60 structures, C-70 and the dual of C-120. For C-120, the eva luation of the (real) determinant of the (complex) matrices that arise was carried out on a personal computer.