The Method of Colored Graphs for Simplifying Expressions with Indices

Abstract Computer algebra is increasingly used in research and applied computations. An example is tensor computations or, in a wide sense, simplification of expressions containing indices. In this paper, the method of colored graphs designed for simplifying abstract expressions containing indices is developed for the case when the indices are of different types. An example of such indices are upper and lower indices in tensor expressions. The proposed approach significantly reduces the number of combinations to be considered when the canonical form of the expression is sought; as a result, the computations are drastically accelerated.