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This work is devoted to the study of application of new topologies in the design of networks-on-chip (NoCs). It is proposed to use two-dimensional optimal circulant topologies for NoC design, and it is developed an optimized routing algorithm with the decreased memory usage. The proposed routing algorithm was compared with Table routing, Clockwise routing, and Adaptive routing algorithms, previously developed for ring circulant topologies, and specialized routing algorithm for multiplicative circulants. The results of synthesis of routers implementing proposed routing algorithms are presented. The cost of ALM and register resources for the implementation of communication subsystems in NoCs with circulant topologies is estimated.

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Recent developments in commutative algebra, linear algebra, and graph theory allow us to approach various issues in several fields. Circulant graphs now have a wider range of practical uses, including as the foundation for optical networks, discrete cellular neural networks, small-world networks, models of chemical reactions, supercomputing and multiprocessor systems. Herein, we are concerned with the decompositions of the bipartite circulant graphs. We propose the Cartesian and tensor product approaches as helping tools for the decompositions. The proposed approaches enable us to decompose the bipartite circulant graphs into many categories of graphs. We consider the use cases of applying the described theory of bipartite circulant graph decomposition to the problems of finding new topologies and deadlock-free routing in them when building supercomputers and networks-on-chip.