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Spectral faux trees

Steve Butler, Elena D’Avanzo, Rachel Heikkinen, Joel Jeffries, Alyssa Kruczek and Harper Niergarth

Vol. 17 (2024), No. 4, 651–668
Abstract

A spectral faux tree with respect to a given matrix is a graph which is not a tree but is cospectral with a tree for the given matrix. We consider the existence of spectral faux trees for several matrices, with emphasis on constructions.

For the Laplacian matrix, there are no spectral faux trees. For the adjacency matrix, almost all trees are cospectral with a faux tree. For the signless Laplacian matrix, spectral faux trees can only exist when the number of vertices is of the form n = 4k. For the normalized adjacency, spectral faux trees exist when the number of vertices, n, is at least 4, and we give an explicit construction for a family whose size grows exponentially with k for n = αk + 1, where α is a fixed integer.

Keywords
cospectral, trees, faux trees, signless Laplacian matrix, normalized adjacency matrix, characteristic polynomials
Mathematical Subject Classification
Primary: 05C50
Milestones
Received: 13 January 2023
Revised: 23 May 2023
Accepted: 27 May 2023
Published: 2 October 2024

Communicated by Glenn Hurlbert
Authors
Steve Butler
Iowa State University
Ames, IA
United States
Elena D’Avanzo
Carleton College
Northfield, MN
United States
Rachel Heikkinen
Augustana College
Rock Island, IL
United States
Joel Jeffries
Iowa State University
Ames, IA
United States
Alyssa Kruczek
Susquehanna University
Selinsgrove, PA
United States
Harper Niergarth
University of Minnesota–Twin Cities
Minneapolis, MN
United States