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On the Simplification of Tensor Expressions

Tensors are mathematical objects that generalize vectors and matrices. They describe geometrical quantities and they are used in various applied settings including mathematical physics. The indicial notation of tensors permits us to write an expression in a compact manner and to use simplifying mathematical operations. In a large number of problems in differential geometry and general relativity, the time consuming and straightforward algebraic manipulation is obviously very important. Thus, tensor computation came into existence and became necessary and desirable at the same time. Over the past 25 years, few algorithms have appeared for simplifying tensor expressions. Among the most important tensor computation systems, we can mention SHEEP, Macsyma !Tensor Package, MathTensor and GRTensorII. Meanwhile, graph theory, which had been lying almost dormant for hundreds of years since the time of Euler, started to explode by the turn of the 20th century. It has now grown into a major discipline in mathematics, which has branched off today in various directions such as coloring problems, Ramsey theory, factorization theory and optimization, with problems permeating into many scientific areas such as physics, chemistry, engineering, psychology, and of course computer science. Investigating some of the tensor computation packages will show that they have some deficiencies. Thus, rather than building a new system and adding more features to it, it was an objective in this thesis to express an efficient algorithms by removing most, if not all, restrictions compared to other packages, using graph theory. A summary of the implementation and the advantages of this system is also included.