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

Abstract

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).

arXiv comment13 pages, 4 figures