Labelling matrices and index matrices of a graph structure

The concept of graph structure was introduced by E. Sampathkumar in 'Generalised Graph Structures', Bull. The concept of graph structure was introduced by E. Sampathkumar in 'Generalised Graph Structures', Bull. Kerala Math. Assoc., Vol. 3, No. 2, Dec 2006, 65-123. Based on the works of Brouwer, Doob and Stewart, R.H. Jeurissen has ('The Incidence Matrix and Labelings of a Graph', J. Combin. Theory, Ser. B30 (1981), 290-301) proved that the collection of all admissible index vectors and the collection of all labellings for $0$ form free $F$-modules ($F$ is a commutative ring). We have obtained similar results on graph structures in a previous paper. In the present paper, we introduce labelling matrices and index matrices of graph structures and prove that the collection of all admissible index matrices and the collection of all labelling matrices for $0$ form free $F$-modules. We also find their ranks in various cases of bipartition and char $F$ (equal to 2 and not equal to 2).