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

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 $A_n$.

💡 Research Summary

The paper investigates the Q‑walk matrix of the Dynkin graph (A_n), denoted (W_Q(A_n)={e_n, Qe_n, Q^2e_n,\dots ,Q^{n-1}e_n}), where (Q=A+D) is the signless Laplacian of the graph. The authors’ primary goal is to determine the exact rank of this matrix and to compute its Smith normal form (SNF).

First, they recall the definition of invariant factors and SNF for an integer matrix, emphasizing that the SNF is a diagonal matrix (\operatorname{diag}(d_1,\dots ,d_r,0,\dots ,0)) with each (d_i) dividing (d_{i+1}). They note that the rank of a walk matrix cannot exceed the number of cells in any equitable partition of the vertex set.

Using the natural symmetry of the path graph (A_n), they introduce two equitable partitions: (\Pi_1) for even (n) (pairs ({1,n},{2,n-1},\dots)) and (\Pi_2) for odd (n) (similar pairing with a central singleton). The characteristic matrices (C_1, C_2) of these partitions lead to reduced divisor matrices (B_1, B_2). Lemma 2.2 shows that the original Q‑walk matrix can be expressed as a walk matrix of a much smaller matrix: \