Directly associate a q-matroid to a q-ary graph

Determine how to construct a q-matroid directly from a q-ary graph, without passing through an incidence matrix, by identifying the appropriate q-analogue of a set of edges and characterizing when that space is a circuit of the associated q-matroid based on whether the corresponding edges form a cycle in the q-ary graph.

Background

The paper defines q-ary graphs whose vertices are one-dimensional subspaces and whose edges are selected two-dimensional subspaces satisfying a closure condition modeled on neighborhoods in ordinary graphs. It then constructs an incidence matrix over an extension field and proves that this matrix represents a q-matroid, with the resulting q-matroid independent of the choices involved up to isomorphism.

The authors explicitly distinguish this incidence-matrix construction from a direct correspondence between q-ary graphs and q-matroids. They identify the missing ingredient as a q-analogue of a set of edges, which should be a subspace, and suggest that its circuit status might be characterized through a suitable notion of cycles in q-ary graphs.