I'm interested in looking for properties of adjacency matrices of trees. On one comment of this question: https://mathoverflow.net/questions/89692/if-graph-is-tree-what-can-be-said-about-its-adjacency-matrix, Some one answered that every adjacency matrix of tree can be written as $N + N^T$ by picking a vertex as root and follow the edges.
It seems reasonable. However, I can't think of a more rigorous proof of it. Can someone help provide some insights? Thanks!
:-) OK, you are looking worthy so I reveal to you a secret which may be not known to these graph theorists who did not study matrix theory: each symmetric square matrix $A=\|a_{ij}\|$ with zeroes on the diagonal (in particular, any adjacency matrix of a finite simple graph) can be represented as $A=N+N^T$, where $N=\|n_{ij}\|$ is nilponent (for each $i,j$ we put $n_{ij}=a_{ij}$, if $i<j$ and $n_{ij}=0$, otherwise).