Linear transformations preserving cyclicity index of graphs and matrices

Cyclicity index of a strongly connected directed graph is equal to the greatest common divisor of the lengths of all its directed cycles. Cyclicity index of a graph is the least common multiple of cyclicity indices of all its strongly connected components. Cyclicity index of a matrix is the cyclicity index of its critical subgraph. It is an important invariant actively used for determination of regular regimes in scheduling and other network problems.
Theory of linear transformations preserving matrix invariants dates back to Frobenius and is actively developing research area. In this talk we discuss linear transformations on different matrix semirings that preserve cyclicity index or just some of its values with particular emphasis on the existence of singular maps.
Based on a series of joint works with A. Guterman, C. Thomassen, and A. Vlasov.