Isospectral flows preserving some centrosymmetric structures

Let GG be a simple undirected graph (no loops, no multiple edges) on n   vertices. Let SnSn be the set of real symmetric matrices of order n  . A matrix A=(ai,j)∈SnA=(ai,j)∈Sn is said to be a matrix on GG if ai,j=0ai,j=0 whenever i≠ji≠j and the vertices i, j   of GG are not joined by an edge of GG. We recall that if F   is a skew-symmetric operator on SnSn, then the solution A(t) ofdAdt=[A,F(A)]A(0)=A0∈Snmaintains the spectrum of A0. The matrix A∈SnA∈Sn is said to be centrosymmetric if JAJ = A, where J   is the matrix with ones on the secondary diagonal and zeros elsewhere. Centrosymmetric matrices are symmetric about the secondary diagonal. Centrosymmetric matrices appear in fields such as finite element analysis. We construct an isospectral flow on a graph GG, with the property that if A0 is centrosymmetric, so is A(t), and discuss the limit of A(t  ) as t→∞t→∞