properties(g k )=T(laws(s j ))

1. Special case Checking the validity of engineering systems through properties of the graph/matroid representations properties(gk)=T(laws(sj)) properties(gk)=T(validity rule(sj)) Mathematical properties of graphs/matroids, such as: planarity, perfectness, connectivity, etc. may present the validity criteria for the represented engineering systems.

2. B C D A 6 1 9 α 10 8 5 2 3 E G 4 7 F 1 9 6 D A C B 2 10 5 8 3 Rα G E F 7 4 RG RE  y x Checking the validity of engineering systems through properties of graph representations: Trusses • A determinate truss isrigid if and only if when doubling each edge in turn in the graph, all the edges can be covered by two edge disjoint spanning trees.

3. Geometric constraint system is valid (well constrained) if and only if its corresponding graph is rigid. A b l1 b d B r a a c D C b B b A r a X l1 d r a C D c Checking the validity of engineering systems through properties of graph representations: Geometric Constraint Systems

4. the turning edges (black) constitute a spanning tree there is one and only one gray vertex in each fundamental circuit. In each fundamental circuit, there is one and only one gray vertex In each fundamental circuit, the levels of the vertex representing a gear wheel and the turning edge incident to it must be identical. Checking the validity of engineering systems through properties of graph representations: Planetary Gear Systems

5. The system of diagonal rods and cables in a square grid is stable if and only if the corresponding bipartite graph is strongly connected. Checking the validity of engineering systems through properties of graph representations: Tensegrity Grids

6. Tensegrity structure is rigid if and only if the corresponding directed matroid is strongly connected. Tensegrity structure can sustain specific external force, if that force is contained in a directed circuit of the corresponding matroid. Checking the validity of engineering systems through properties of graph representations: Tensegrity Structures