The smith normal form of the walk matrix of the Dynkin graph $A_n$

Abstract

In this paper, we give the rank of the walk matrix of the Dynkin graph $A_n$, and prove that its Smith normal form is $$\diag(\underbrace{1,\ldots,1}_{\lceil\frac{n}{2}\rceil},0,\ldots,0).$$

14 pages