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

TL;DR

This paper demonstrates that transition matrices from continuous vertex maps on trees are similar over certain fields and have a specific characteristic polynomial, revealing uniform algebraic properties across all such maps.

Contribution

It establishes the similarity and characteristic polynomial form of transition matrices for all continuous vertex maps on oriented trees, both oriented and unoriented.

Findings

Transition matrices are similar over ield and ield for all maps.

Characteristic polynomial is orm or all such matrices.

Coefficients are all odd integers when considered over ield.

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).