论文标题
在dynkin图的步行矩阵上$ d_n $
On the walk matrix of the Dynkin graph $D_n$
论文作者
论文摘要
令$ w（d_n）$表示dynkin图$ d_n $的步行矩阵，这是从订单$ n-1 $的路径获得的树，通过在第二个顶点添加吊坠边缘。我们证明$ \ text {rank} \，w（d_n）= n-2 $，如果$ 4 \ mid n $和$ \ text {rank} \，w（d_n）= n-1 $否则。此外，我们证明$ W（d_n）$的史密斯正常形式是$$ \ text {diag} [\ Underbrace {1,1，\ ldots，1} _ {\ lceil \ frac {n} {2} {2} \ rcei l}，\ UnderBrace {2,2，\ ldots，2} _ {\ lfloor \ frac {n} {2} {2} \ rfloor-1}，0] $$ $ 4 \ nmid n $。这证实了[W.Wang，F.liu，W.Wang，几乎可控图的广义频谱特征，欧洲J. Combin。，96（2021）：103348]。
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 $$\text{diag}[\underbrace{1,1,\ldots,1}_{\lceil\frac{n}{2}\rceil},\underbrace{2,2,\ldots,2}_{\lfloor\frac{n}{2}\rfloor-1},0]$$ 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].
