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

TL;DR

This paper determines the rank and Smith normal form of the walk matrix for the Dynkin graph $A_n$, revealing a specific diagonal structure with implications for graph theory and algebraic properties.

Contribution

It provides the exact Smith normal form of the walk matrix for $A_n$, a result not previously established.

Findings

Rank of the walk matrix is $ ceil n/2 ceil$.

Smith normal form has $ ceil n/2 ceil$ ones followed by zeros.

The structure aids in understanding algebraic properties of Dynkin graphs.

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 $$\diag(\underbrace{1,\ldots,1}_{\lceil\frac{n}{2}\rceil},0,\ldots,0).$$