@article{Dai2026, 
author = {Li Dai},
title = {Laplacian controllability analysis of chain graph based on minimal perfect critical vertex set},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {3},
pages = {7766-7778},
keywords = {minimal perfect critical vertex set, chain graph, Laplacian controllability, minimum leader set, subgraph structure},
url = {https://www.sciopen.com/article/10.3934/math.2026319},
doi = {10.3934/math.2026319},
abstract = {The aim is to find all of the minimum leader set for chain graphs. By analyzing the subgraph structure rather than the eigenvectors, minimal perfect critical vertex sets (MPCS) of chain graph are found. It is proved that there is one and only one MPCS in chain graph (CG) (1, 1, ⋯, 1; 1, 1, ⋯, 1), and it is a 4-MPCS. Based on this, all the minimum leader sets of the chain graph are given.}
}