树上顶点映射的相似平方{-1，0，1}-矩阵类
On the Class of Similar Square {-1,0,1}-Matrices Arising from Vertex maps on Trees
摘要：
令$n \ge 2$为整数。本文证明了在具有$n+1$顶点的{ \it all}定向树上的所有连续{ \it顶点映射}的所有实数域$\mathcal R$上（分别在两个元素的有限域$\mathcal Z_2$上）的{\it定向}转移阵在$\mathcal R$上（分别在$\mathcal Z_2$上）彼此相似，并具有特征多项式$\sum_{k=0}^n x^k$ .因此，在具有$n+1$顶点的{\it all}定向树上的所有连续{ \it顶点映射}的域$Z_2$上的{ \it无定向}转移阵在$\mathcal Z_2$上彼此相似，并且具有特征多项式$\sum_{k=0}^n x^k$。因此，当在域$\mathcal R$上考虑时，这些{ \it无向}转移阵的特征多项式的系数都是奇整数（因此非零）。
英文摘要：
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).
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