On the walk matrix of the Dynkin graph $D_n$

Abstract

Let $W(D_n)$ denote the walk matrix of the Dynkin graph $D_n$, a tree obtained from the path of order $n-1$ by adding a pendant edge at the second vertex. We prove that $\text{rank}W(D_n)=n-2$ if $4\mid n$ and $\text{rank}W(D_n)=n-1$ otherwise. Furthermore, we prove that the Smith normal form of $W(D_n)$ is $$\text{diag}[\underset{\lceil \frac{n}{2}\rceil }{\underbrace{1,1,\dots ,1}},\underset{\lfloor \frac{n}{2}\rfloor -1}{\underbrace{2,2,\dots ,2}},0]$$ when $4\nmid n$. This confirms a recent conjecture in [W.Wang, F.Liu, W.Wang, Generalized spectral characterizations of almost controllable graphs, European J. Combin., 96(2021):103348].

11pages, 1figure