Distinct eigenvalues are realizable with generic eigenvectors

Levene, Rupert H. (author), Oblak, Polona (author), Šmigoc, Helena (author)

Motivated by applications in matrix constructions used in the inverse eigenvalue problem for graphs, we study a concept of genericity for eigenvectors associated with a given spectrum and a connected graph. This concept generalizes the established notion of a nowhere-zero eigenbasis. Given any simple graph G on n vertices and any spectrum with no multiple eigenvalues, we show that the family of eigenbases for symmetric matrices with this graph and spectrum is generic, strengthening a result of Monfared and Shader. We illustrate applications of this result by constructing new achievable ordered multiplicity lists for partial joins of graphs and providing several families of joins of graphs that are realizable by a matrix with only two distinct eigenvalues.

• Language: English
• Keywords: symmetric matrix, join of graphs, inverse eigenvalue problem, minimal number of distinct eigenvalues
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FRI - Faculty of Computer and Information Science
• Publication date: 01.01.2023
• Year: 2023
• Number of pages: Str. 1-15
• Numbering: Vol. , no.
• PID: 20.500.12556/RUL-148518
• UDC: 51
• ISSN on article: 0308-1087
• DOI: 10.1080/03081087.2023.2232090
• COBISS.SI-ID: 162220803
• Publication date in RUL: 25.08.2023

Record is a part of a journal

• Title: Linear and multilinear algebra
• Shortened title: Linear multilinear algebra
• Publisher: Taylor & Francis
• ISSN: 0308-1087
• COBISS.SI-ID: 25872128

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: simetrična matrika, spoj grafov, inverzni problem lastnih vrednosti, najmanjše število različnih lastnih vrednosti

Projects

• Funder: ARRS - Slovenian Research Agency
• Project number: P1-0222
• Name: Algebra, teorija operatorjev in finančna matematika