T-Type Digraphs via Kronecker Canonical Form

• Published in: Journal of systems science and complexity Vol. 38; no. 3; pp. 1259 - 1280
• Main Authors: Chen, Sheng, Hu, Haofei
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2025; Springer Nature B.V
• Subjects: Algorithms; Canonical forms; Complex Systems; Control; Graph theory; Mathematics; Mathematics and Statistics; Mathematics of Computing; Operations Research/Decision Theory; Statistics; Systems Theory; Tensors; tree; Kronecker canonical form; incidence matrix; unimodular matrix; Digraph; tensor product
• ISSN: 1009-6124; 1559-7067
• DOI: 10.1007/s11424-024-3301-2

In this paper, the authors aim to study Kronecker canonical form theory for T-type digraphs, which can be used to construct trees by tensor product with some directed paths. Firstly, the authors show that some bicyclic digraphs and multicyclic digraphs are T-type digraphs. Secondly, the authors provide a characterization for T-type digraphs by their Kronecker canonical form. Moreover, the authors present an algorithm for computing the Kronecker canonical form, which can be used to determine whether or not a digraph is a T-type digraph. Lastly, for a class of T-type digraphs, the authors show that their incidence matrix pair can be transformed into Kronecker canonical form using unimodular matrices. The authors also present an algorithm related to finding such unimodular matrices.