On the Class of Similar Square {-1,0,1}-Matrices Arising from Vertex maps on Trees

Let $n \ge 2$ be an integer. In this note, we show that the {\it oriented} transition matrices over the field $\mathcal R$ of all real numbers (over the finite field $\mathcal Z_2$ of two elements respectively) of all continuous {\it vertex maps} on {\it all} oriented trees with $n+1$ vertices are similar to one another over $\mathcal R$ (over $\mathcal Z_2$ respectively) and have characteristic polynomial $\sum_{k=0}^n x^k$. Consequently, the {\it unoriented} transition matrices over the field $Z_2$ of all continuous {\it vertex maps} on {\it all} oriented trees with $n+1$ vertices are similar to one another over $\mathcal Z_2$ and have characteristic polynomial $\sum_{k=0}^n x^k$. Therefore, the coefficients of the characteristic polynomials of these {\it unoriented} transition matrices, when considered over the field $\mathcal R$, are all odd integers (and hence nonzero).