Modular topological structures, commonly known as Assur groups, have been an important subject of research in machine theory over the past several decades. In this paper, some unique engineering properties which appear in this type of structures and only in them are exposed. The concept of Assur groups is reformulated in terms of graphs and named Assur Graphs (AGs). The graph representation enables to present the inter-connectivity between all the elements of the AG, which is the fundamental characteristic from which all its unique properties derive. This inter-connectivity leads to unique properties in three main engineering aspects of structures: kinematics, statics and singularity positions. These properties can be exploited for better analysis and synthesis of structures such as mechanisms, robots and trusses.

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