The Index Of The 0-1 Matrix

Let Mn{0,1} be the set of all n-order 0-1 matrices.If Ai ?Mn{0,1},i =0,1,2,...,k,and Ak+1(?)Mn{0,1},k is called the index of A,it is also called matrix index,which is represented by ?(A)= k.When k can take any positive integer,it is written as ?(A)=?.The main problem solved by the paper is to find f(n)=max{?(A)|A E Mn{0,1},?(A)<?} and the maximum number of elements 1 in matrix A0 and the characteristics of A0 when f(n)=?(A0).When n<7,all 0-1 matrices can be easily obtained by MATLAB,and can be classified according to whether the matrix index is the same.When n?7,the MATLAB operation takes too long,so the directed graph is used to study the properties of the matrix.First,the directed graph of the higher-order matrix is analogized from the directed graph of the low-order matrix.The transformation of the graph makes the matrix corresponding to the directed graph increase the value of the matrix index.At this time,the directed graph corresponding to the matrix corresponding to the maximum value g(n)is g(p,q).Next,we discuss the standard subblock of triangular block under the permutation similarity of matrix A,and ?(A)?g(n)(n?7),f(n)=g(n).which is(?)A0?Mn{0,1},A0i?Mn{0,1},i = 0,1,2,...,f(n),(n ? 7),but A0f(n)+1(?)Mn{0,1},and matrix A0 is similar to matrix where B is a matrix with only one element being 1 and the rest being 0,with max#{1}=n+1.In the process of solving the problem,some conclusions about the index of the symmetric 0-1 matrix and the index of the lower triangular matrix are also obtained.