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

Wei Wang

doi:10.1016/j.laa.2023.04.021

paper

The Smith normal form of the walk matrix of the Dynkin graph $D_{n}$ for $n \equiv 0 (mod 4)$

arXiv:2212.05456 · doi:10.1016/j.laa.2023.04.021

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 $diag [\frac{n}{2} - 1 1, 1, \dots, 1, \frac{n}{2} - 1 2, 2, \dots, 2, 0, 0]$ when $n \equiv 0 (mod 4)$ . 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].

12 pages,1 figure

The Smith normal form of the walk matrix of the Dynkin graph Dn​ for n≡0(mod4)

Abstract

References in corpus (2)

The Smith normal form of the walk matrix of the Dynkin graph $D_{n}$ for $n \equiv 0 (mod 4)$