On linear transformations preserving cyclicity index of nonnegative matrices

The cyclicity index of a strongly connected directed graph is the greatest common divisor of all its directed cycles and the cyclicity index of an arbitrary directed graph is the least common multiple of the cyclicity indices of all its maximal strongly connected subgraphs. The cyclicity index of a matrix is equal to the cyclicity index of its critical subgraph, namely, the subgraph of the adjacent graph consisting of all cycles with the maximal average weight. In this paper, we consider surjective linear transformations of non-negative and integer non-negative matrices preserving the cyclicity index. We obtain a complete characterization of such maps and prove that they are automatically injective.