The Nonlinear matrix Equation Xm = A and its Applications to Graph Theory

Nonlinear matrix equation widely used in the control theory.When restricted to integer rings ,the integer matrix and the finite directed graph is corresponding.This paper made the integer root of matrix equation applied to the graph theory, and get several propositions about existence of the underlying graph(root graph) of special graphs. And we obtained the algebraic approach on how to find the adjacency matrix of the underlying graph(root graph). Finally, we also determine the precise number of solutions in various cases.