  The G-Wishart distribution is an essential component for the Bayesian
analysis of Gaussian graphical models as the conjugate prior for the precision
matrix. Evaluating the marginal likelihood of such models usually requires
computing high-dimensional integrals to determine the G-Wishart normalising
constant. Closed-form results are known for decomposable or chordal graphs,
while an explicit representation as a formal series expansion has been derived
recently for general graphs. The nested infinite sums, however, do not lend
themselves to computation, remaining of limited practical value. Borrowing
techniques from random matrix theory and Fourier analysis, we provide novel
exact results well suited to the numerical evaluation of the normalising
constant for a large class of graphs beyond chordal graphs. Furthermore, they
open new possibilities for developing more efficient sampling schemes for
Bayesian inference of Gaussian graphical models.
