On The Stable Index Of Several Types Of 0-1 Matrices

Let Mn{0,1} be the set of all n-order 0-1 matrices.If A1?Mn?0,1?,i=0,1,2,…,k,but Ak+1(?){0,1},k is called the stable index of A,which is represented by s(A)=k.When k can be taken any positive integer,it is written as s(A)=x.There are three main conclusions for symmetric 0-1 matrices and triangular 0-1 matrices:the stable index of a principal minor is greater than or equal to its own stable index;The stable index has some monotonic property,if all the elements of A do not exceed the corresponding elements of B,then the stable index of A is not less than the stable index of B.If the stable index of a symmetric 0-1 matrix is finite,it must be 1.If the stable index of a triangular 0-1 matrix is finite,it should not be greater than the order of the matrix.Since the properties of the graph can be characterized by the algebraic properties of the matrix(including eigenvalues,the rank of matrix,matrix stable index,etc.),one of the conclusion:the square sum and cubic sum of the eigenvalues of the symmetric 0-1 matrix are two times of the number of sides and six times the number of triangles in the corresponding simple graph,respectively.For the lower order(the order m no more than 5)central symmetric 0-1 matrices,some sufficient conditions for the determination of its stable index are obtained.For example,denote Pm is a matrix whose elements are in {0,1,2},if some diagonal element is 2 or there are at least 2 elements is 2 in the minor diagonal,then the stable index of the corresponding of the central symmetric 0-1 matrix is 1.For higher order central symmetric 0-1 matrices,applying permutation matrix and similarity transformation to reduce the dimension,so as to reduce the computational complexity.However,as the dimension of the matrix increases,its stable index usually does not become larger(except the infinity of circumstances).In addition,the upper bound for the stable index of even order and odd order central symmetric 0-1 matrices are min{s(A4)}and min{s(A5)},respectively.