The Smith normal form of the Q-walk matrix of the Dynkin graph $A_n$

Abstract

In this paper, we give an explicit formula for the rank of the $Q$-walk matrix of the Dynkin graph $A_n$. Moreover, we prove that its Smith normal form is $$\mathrm{diag}\left(\underset{r=\lceil \frac{n}{2}\rceil }{\underbrace{1,2,2,...,2}},0,...,0\right),$$ where $r$ is the rank of the $Q$-walk matrix $W_Q\left(A_n\right)$ of the Dynkin graph $A_n$.

All commets are welcome