This paper gives a class of multilinear uncertain polynomials to which the Edge Theorem is applicable. In the paper, a notion of "symmetric uncertainty structure" is introduced. The main result is as follows : If a multilinear uncertain polynomial has a symmetric uncertainty structure and the uncertain parameters are nonoverlapping, then the corresponding value sets will be convex polygons. This means that the use of the zero exclusion condition to test the robust stability of such uncertain polynomials is feasible since we need only to calculate the extreme points of the convex polygons. Furthermore, a version of the Edge Theorem can be, used to test the robust stability of such uncertain polynomials.