Eigenvalue Problems: Isospectrality and Inverse Eigenvalue Problems
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 $G$.