Computation of spanning tree generators of fullerenes

Abstract

Laplacians of fullerenes are computed as generators of the number of spanning trees which find applications in the computation of the magnetic properties of fullerenes. The computed Laplacian polynominals which enumerate the spanning trees of various subgraphs of fullerenes are used for obtaining the analytical forms for the first nine coefficients of fullerenes. It is shown that the eighth and higher coefficients depend on the structure rather than purely on the number of vertices although, for fullerenes containing isolated pentagons, coefficients up to the twelfth do not exhibit structural dependence. All computations were made in quadruple precision arithmetic on an IBM RS/6000 580 workstation.

On Laplacian Eigenvalues of Wheel Graphs

“…The above result directly follows from the previous results [11,12], which show that the sum of the Laplacian spectra equals the sum of the degrees of all the vertices of the graph, which can be readily seen to be 4n for W n+1 .…”

“…In 1995, Balasubramanian [11] found the Laplacian of fullerenes C 42 − C 90 (see Section 1.2) using high-precision computational algorithms. In [12], the same author computed the Laplacians of fullerenes as the generators of the number of spanning trees, which find applications in the computation of the magnetic properties of fullerenes. In 2001, the authors of [13] investigated cyclicity in four types of polycyclic graphs, including five-vertex graphs with a five-cycle, Schlegel graphs depicting platonic solids, buckminsterfullerene isomers, and C 70 isomers, using the distance-related and resistance distance-related indices.…”