  Passerini and Severini have recently shown that the Braunstein-Ghosh-Severini
(BGS) entropy S(Gamma) = -Tr[rho(Gamma) log rho(Gamma)] of a certain density
matrix rho(Gamma) naturally associated to a graph Gamma, is maximized, among
all graphs with a fixed number of links and nodes, by regular graphs. We ask if
this result can play a role in quantum gravity, and be related to the apparent
regularity of the physical geometry of space. We show that in Loop Quantum
Gravity the matrix rho(Gamma) is precisely the Hamiltonian operator (suitably
normalized) of a non-relativistic quantum particle interacting with the quantum
gravitational field, if we restrict elementary area and volume eigenvalues to a
fixed value. This operator provides a spectral characterization of the physical
geometry, and can be interpreted as a state describing the spectral information
about the geometry available when geometry is measured by its physical
interaction with matter. It is then tempting to interpret its BGS entropy
S(Gamma) as a genuine physical entropy: we discuss the appeal and the
difficulties of this interpretation.
