  The recursion and path-integral methods are applied to analytically study the
electronic structure of a neutral $C_{60}$ molecule. We employ a tight-binding
Hamiltonian which considers both the $s$ and $p$ valence electrons of carbon.
From the recursion method, we obtain closed-form {\it analytic} expressions for
the $\pi$ and $\sigma$ eigenvalues and eigenfunctions, including the highest
occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital
(LUMO) states, and the Green's functions. We also present the local densities
of states around several ring clusters, which can be probed experimentally by
using, for instance, a scanning tunneling microscope. {}From a path-integral
method, identical results for the energy spectrum are also derived. In
addition, the local density of states on one carbon atom is obtained; from this
we can derive the degree of degeneracy of the energy levels.
