Abstract: We introduce the stability coefficients and stable sets of complex polynomials. We define matrix graphs and the construction structures set generated by a matrix graph. We introduce matrix networks linked to graph theory. We prove that any n-tuple of commuting contractions of matrix networks satisfies the von Neumann’s inequality. We define complex polynomials over N which don’t have any positive integer roots but which have matrix roots with positive integers as entries. We show that these matrix roots are construction structures of matrix solutions of Diophantine equations. In particular, we show that the Diophantine equation X_1^n+⋯+X_m^n=X_(m+1)^n+n,m ≥ 2, admits an infinite number of matrix solutions with positive integers as entries.