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

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 $$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(,0,\mathrm{.}\mathrm{.}\mathrm{.},0),$$ where $r$ is the rank of the $Q$-walk matrix $W_Q\left(A_n\right)$ of the Dynkin graph $A_n$.