The Smith normal form of the walk matrix of the Dynkin graph $D_n$ for $n\equiv 0\pmod{4}$

TL;DR

This paper determines the Smith normal form of the walk matrix for the Dynkin graph $D_n$ when $n$ is divisible by 4, confirming a previous conjecture and providing explicit matrix invariants.

Contribution

It explicitly computes the Smith normal form of the walk matrix for $D_n$ graphs with $n mod 4=0$, answering an open question in the literature.

Findings

Smith normal form of $W(D_n)$ is diagonal with specified entries

Confirmed the conjecture posed in prior work

Provides explicit matrix invariants for $D_n$ when $nmod 4=0$

Abstract

Let $W(D_n)$ denote the walk matrix of the Dynkin graph $D_n$. We prove that the Smith normal form of $W(D_n)$ is $$\textup{diag}[\underbrace{1,1,\ldots,1}_{\frac{n}{2}-1},\underbrace{2,2,\ldots,2}_{\frac{n}{2}-1},0,0]$$ when $n\equiv 0\pmod{4}$. This gives an affirmative answer to a question in [W. Wang, C. Wang, S. Guo, On the walk matrix of the Dynkin graph $D_n$, Linear Algebra Appl. 653 (2022) 193--206].