  Gaussian graphical models play an important role in various areas such as
genetics, finance, statistical physics and others. They are a powerful
modelling tool which allows one to describe the relationships among the
variables of interest. From the Bayesian perspective, there are two sources of
randomness: one is related to the multivariate distribution and the quantities
that may parametrise the model, the other has to do with the underlying graph,
$G$, equivalent to describing the conditional independence structure of the
model under consideration. In this paper, we propose a prior on G based on two
loss components. One considers the loss in information one would incur in
selecting the wrong graph, while the second penalises for large number of
edges, favouring sparsity. We illustrate the prior on simulated data and on
real datasets, and compare the results with other priors on $G$ used in the
literature. Moreover, we present a default choice of the prior as well as
discuss how it can be calibrated so as to reflect available prior information.
