About one heritage character in cyclic semigroups of graphs

This paper is devoted to describing the character of loop existing at the given vertex subset in a cyclic semigroup of digraphs named as loop-character. Some attained bounds and formulae for the loop-character index are obtained. They are expressed in terms of circuit lengths in a digraph, which is the generator of the cyclic semigroup. An example showing the exactness of the bounds is given. The results can be used for estimating exponents of wide range primitive digraph systems.