Campus Units
Mathematics
Document Type
Article
Publication Version
Accepted Manuscript
Publication Date
5-2020
Journal or Book Title
Journal of Combinatorial Theory, Series B
Volume
142
First Page
276
Last Page
306
DOI
10.1016/j.jctb.2019.10.005
Abstract
The inverse eigenvalue problem of a given graph G is to determine all possible spectra of real symmetric matrices whose off-diagonal entries are governed by the adjacencies in G. Barrett et al. introduced the Strong Spectral Property (SSP) and the Strong Multiplicity Property (SMP) in [Generalizations of the Strong Arnold Property and the minimum number of distinct eigenvalues of a graph. Electron. J. Combin., 2017]. In that paper it was shown that if a graph has a matrix with the SSP (or the SMP) then a supergraph has a matrix with the same spectrum (or ordered multiplicity list) augmented with simple eigenvalues if necessary, that is, subgraph monotonicity. In this paper we extend this to a form of minor monotonicity, with restrictions on where the new eigenvalues appear. These ideas are applied to solve the inverse eigenvalue problem for all graphs of order five, and to characterize forbidden minors of graphs having at most one multiple eigenvalue.
Creative Commons License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.
Copyright Owner
Elsevier Inc.
Copyright Date
2019
Language
en
File Format
application/pdf
Recommended Citation
Barrett, Wayne; Butler, Steve; Fallat, Shaun M.; Hall, H. Tracy; Hogben, Leslie; Lin, Jephian C.-H.; Shader, Bryan L.; and Young, Michael, "The inverse eigenvalue problem of a graph: Multiplicities and minors" (2020). Mathematics Publications. 199.
https://lib.dr.iastate.edu/math_pubs/199
Comments
This is a manuscript of an article published as Barrett, Wayne, Steve Butler, Shaun M. Fallat, H. Tracy Hall, Leslie Hogben, Jephian C-H. Lin, Bryan L. Shader, and Michael Young. 142 "The inverse eigenvalue problem of a graph: Multiplicities and minors." Journal of Combinatorial Theory, Series B (2020): 276-306. DOI: 10.1016/j.jctb.2019.10.005. Posted with permission.