Isospectral matrix flows are dynamical systems evolving on the space of matrices, which have specific properties that their solutions maintain the eigenvalues of the initial matrix. These flows are of particular interest to numerical analysis because of their connection to algorithms where the eigenvalues of a certain matrix remain fixed throughout the computation, such as QR algorithm. Such isospectral flows appear to be a useful tool for solving inverse eigenvalue problem.

Due to their importance in engineering applications, inverse eigenvalue problems have received considerable attention. Many inverse eigenvalue problems reduce to the construction of a matrix with a prescribed structure (e.g., tridiagonal, pentadiagonal, Toeplitz, etc.) and spectral data.

Matrix analysis is the language of discrete systems and the language of graph theory that is needed to analyse them. Let $S_n$ be the set of real symmetric matrices of order $n$. The matrix $A = (a_{ij}) \in S$ is said to lie on a strict undirected graph $G(V, E)$ (no loops, no multiple edges) if $a_{ij} = 0$ ($i \neq j$) whenever $(i, j)$ is not in $E$. First, we consider an isospectral flow on a given graph and show that how this flow maintains some properties on a graph. Then we show that for each set of spectral data and each Tree $G(V,E)$ on $n$ vertices there exists an $n \times n$ real symmetric matrix $A$ with the same zero pattern of the adjacency matrix of $G$.