On Graphs and Structural Formulas of the Mechanisms Theory

Structural formulas in the theory of mechanisms are formulas expressing the number of degrees of freedom of a device in terms of the numbers of its links and kinematic pairs. It is well known that they are not always true. Mathematical graph theory helps to understand this phenomenon. The validity of structural formulas in the case of generic frameworks is completely determined by their structure, described by graphs. The present paper considers two models of planar frameworks with rotational pairs. The first model is a construction made up of straight rods (levers) bearing hinges at the ends. Such devices are naturally associated with a graph \( G \) with vertices corresponding to hinges and edges corresponding to levers. In the theory of mechanisms, it is customary to consider another graph \( \cal G \) whose vertices correspond to links and the edges correspond to kinematic pairs. It turns out that the use of the graph \( G \) to describe the structure both in the first model and in the second one, which contains all planar constructions with rotational pairs, is preferable to the graph \( \cal G \). In particular, it allows one to provide a criterion for the applicability of structural formulas for generic devices of a given structure.