Linear Transformations Preserving the Minimal Values of the Cyclicity Index of Tropical Matrices

The cyclicity index of a directed graph is defined as the least common multiple of the cyclicity indices of all its strongly connected components, and the cyclicity index of a strongly connected directed graph is equal to the greatest common divisor of the lengths of all its directed cycles. The cyclicity index of a tropical matrix is the cyclicity index of its critical subgraph, i.e., the subgraph of the adjacency graph consisting of all cycles with the largest average weight. This paper considers linear transformations of tropical matrices that preserve two values of the cyclicity index, 1 and 2. A complete characterization of such transformations is obtained. To this end, it is proved that the values 1 and 2 of the cyclicity index are preserved if and only if all its values are preserved. It is shown that there are mappings of another type that preserve only one value of the cyclicity index.