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

TL;DR

This paper derives an explicit formula for the rank and Smith normal form of the Q-walk matrix associated with the Dynkin graph A_n, revealing its algebraic structure and rank properties.

Contribution

It provides a new explicit formula for the rank and Smith normal form of the Q-walk matrix of A_n, advancing understanding of its algebraic properties.

Findings

The rank of the Q-walk matrix is loor{n/2}ul

The Smith normal form has diagonal entries with loor{n/2}ul 1s and 2s, followed by zeros

The explicit formula clarifies the algebraic structure of the Q-walk matrix

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 $A_n$.