  A comprehensive graph theoretical and finite geometrical study of the
commutation relations between the generalized Pauli operators of N-qudits is
performed in which vertices/points correspond to the operators and edges/lines
join commuting pairs of them. As per two-qubits, all basic properties and
partitionings of the corresponding Pauli graph are embodied in the geometry of
the generalized quadrangle of order two. Here, one identifies the operators
with the points of the quadrangle and groups of maximally commuting subsets of
the operators with the lines of the quadrangle. The three basic partitionings
are (a) a pencil of lines and a cube, (b) a Mermin's array and a bipartite-part
and (c) a maximum independent set and the Petersen graph. These factorizations
stem naturally from the existence of three distinct geometric hyperplanes of
the quadrangle, namely a set of points collinear with a given point, a grid and
an ovoid, which answer to three distinguished subsets of the Pauli graph,
namely a set of six operators commuting with a given one, a Mermin's square,
and set of five mutually non-commuting operators, respectively. The generalized
Pauli graph for multiple qubits is found to follow from symplectic polar spaces
of order two, where maximal totally isotropic subspaces stand for maximal
subsets of mutually commuting operators. The substructure of the (strongly
regular) N-qubit Pauli graph is shown to be pseudo-geometric, i. e., isomorphic
to a graph of a partial geometry. Finally, the (not strongly regular) Pauli
graph of a two-qutrit system is introduced; here it turns out more convenient
to deal with its dual in order to see all the parallels with the two-qubit case
and its surmised relation with the generalized quadrangle Q(4, 3), the dual
ofW(3).
