Properties of paths in graphs and multigraphs

The length of the shortest full path in a strongly connected orgraph with $n$ vertices is estimated. The exponent of the vertex incedent matrix of the graph with self-loop is estimated too. A polynomial estimate is obtained for the subexponent of the system of vertex incedent matrices of $n$-vertex graphs whose union is a strongly connected graph. These results can be used to investigate essential variables of the coordinate functions which define compositions of transformations of the set of finite words.