Bordering of symmetric matrices and an application to the minimum number of distinct eigenvalues for the join of graphs

Aida Abiad,Shaun M Fallat,Mark Kempton,Rupert H Levene,Polona Oblak,Helena Šmigoc,Michael Tait,Kevin N Vander Meulen

doi:10.1016/j.laa.2023.09.013

Bordering of symmetric matrices and an application to the minimum number of distinct eigenvalues for the join of graphs

Aida Abiad, Shaun M Fallat + Show 6 more

Open Access

https://doi.org/10.1016/j.laa.2023.09.013

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Journal: Linear Algebra and Its Applications	Publication Date: Sep 17, 2023
License type: cc-by-nc-nd

Affiliation: Villanova University, Eindhoven University of Technology, Ghent University, University of Regina, University College Dublin, Redeemer University College, Institute of Mathematics, Physics, and Mechanics, University of Ljubljana, Brigham Young University

#Minimum Number Of Distinct Eigenvalues #Eigenvalues Of Graph + Show 8 more

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Abstract

An important facet of the inverse eigenvalue problem for graphs is to determine the minimum number of distinct eigenvalues of a particular graph. We resolve this question for the join of a connected graph with a path. We then focus on bordering a matrix and attempt to control the change in the number of distinct eigenvalues induced by this operation. By applying bordering techniques to the join of graphs, we obtain numerous results on the nature of the minimum number of distinct eigenvalues as vertices are joined to a fixed graph.

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