Subdivision and graph eigenvalues

Kumar, Hitesh (author), Mohar, Bojan (author), Pragada, Shivaramakrishna (author), Zhan, Hanmeng (author)

This paper investigates the asymptotic nature of graph spectra when some edges of a graph are subdivided sufficiently many times. In the special case where all edges of a graph are subdivided, we find the exact limits of the $k$-th largest and $k$-th smallest eigenvalues for any fixed $k$. Given a graph, we show that after subdividing sufficiently many times, all but $O(1)$ eigenvalues of the new graph will lie in the interval $[-2, 2]$. We examine the eigenvalues of the new graph outside this interval, and we prove several results that might be of independent interest.

• Language: English
• Keywords: graph eigenvalues, subdivision, adjacency matrix, interval multiplicity, unimodality
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.04.2025
• Year: 2025
• Number of pages: Str. 336-355
• Numbering: Vol. 710
• PID: 20.500.12556/RUL-167489
• UDC: 519.17
• ISSN on article: 0024-3795
• DOI: 10.1016/j.laa.2025.01.044
• COBISS.SI-ID: 227166211
• Publication date in RUL: 24.02.2025

Record is a part of a journal

• Title: Linear algebra and its applications
• Shortened title: Linear algebra appl.
• Publisher: Elsevier
• ISSN: 0024-3795
• COBISS.SI-ID: 1119247

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.