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

Jia yaning, Shengyong Pan

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 $A_n$.

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 -walk matrix of the Dynkin graph . Moreover, we prove that its Smith normal form is where is the rank of the -walk matrix of the Dynkin graph .

Paper Structure

This paper contains 6 sections, 14 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Smith normal form of $W_Q\left( A_n \right)$ for $n$ is even
  5. Smith normal form of $W_Q\left( A_n \right)$ for $n$ is odd
  6. Concluding remarks

Key Result

Lemma 2.1

The matrices $W_Q\left(A_n \right)$ and $\overline{W_Q\left( A_n \right) }\oplus O_{n-r}$ have the same Smith normal form, where $r=\lceil \frac{n}{2} \rceil$.

Theorems & Definitions (27)

  • Example 2.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 17 more