This article gives a survey of all results on the power graphs of groups and semigroups obtained in the literature. Various conjectures due to other authors, questions and open problems are also included.

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The endomorphism monoids of graphs have been actively investigated. They are convenient tools expressing asymmetries of the graphs. One of the most important classes of graphs considered in this framework is that of Cayley graphs. Our paper proposes a new method of using Cayley graphs for classification of data. We give a survey of recent results devoted to the Cayley graphs also involving their endomorphism monoids.

The outcomes obtained can be used to determine the optimal sequences of tests for each individual cost-function by following the ODPF procedure. The results show that the best single Ewing test for diagnosing CAN is the deep breathing heart rate variation test. Optimal sequences found for the cost-function equal to the number of tests guarantee that the best accuracy is achieved after any number of tests and provide an improvement in comparison with the previous ordering of tests or a random sequence.

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