# On linear structure of non-commutative operator graphs

## TL;DR

This paper investigates the linear structure of non-commutative operator graphs generated by group actions, providing explicit formulas and descriptions for graphs associated with the circle and Heisenberg-Weyl groups.

## Contribution

It offers a new description of the linear structure of certain non-commutative operator graphs, including explicit formulas for their dimensions.

## Key findings

- Graphs generated by the circle group have unitary generators permuting basis vectors.
- Explicit formula for the dimension of graphs generated by the Heisenberg-Weyl group.
- New characterization of the linear structure of these operator graphs.

## Abstract

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.