  We continue the study of non-commutative operator graphs generated by
resolutions of identity covariant with respect to unitary actions of the circle
group and the Heisenber-Weyl group as well. It is shown that the graphs
generated by the circle group has the system of unitary generators fulfilling
permutations of basis vectors. For the graph generated by the Heisenberg-Weyl
group the explicit formula for a dimension is given. Thus, we found a new
description of the linear structure for the operator graphs introduced in our
previous works.
