Lin, J. C.-H., & Shirazi, M. N. (2024). Inverse Fiedler vector problem of a graph. arXiv preprint arXiv:2410.09736.
This paper investigates the inverse Fiedler vector problem, aiming to determine all possible Fiedler vectors for a given graph across different weight assignments of its Laplacian matrix. The authors specifically focus on characterizing possible Fiedler vectors for trees and cycles.
The authors utilize algebraic and combinatorial approaches to analyze the properties of weighted Laplacian matrices and their eigenvectors. They leverage concepts like the Perron-Frobenius theorem, Cauchy interlacing theorem, bottleneck matrices, and Dirichlet matrices to establish relationships between Fiedler vectors and graph structure.
The study provides a theoretical framework for understanding the relationship between the structure of trees and cycles and their potential Fiedler vectors. The findings have implications for applications of Fiedler vectors in graph partitioning, drawing, spectral clustering, and identifying characteristic sets in trees.
This research contributes to the field of spectral graph theory by providing a deeper understanding of the inverse Fiedler vector problem. The characterization of Fiedler vectors for trees and cycles offers valuable insights into the spectral properties of these fundamental graph classes.
The paper primarily focuses on trees and cycles. Further research could explore the inverse Fiedler vector problem for more general graph classes. Additionally, investigating the multiplicity of the algebraic connectivity and its impact on the set of possible Fiedler vectors could be a promising research direction.
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by Jephian C.-H... at arxiv.org 10-15-2024
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