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Spectral faux
trees
Steve Butler, Elena D’Avanzo, Rachel Heikkinen, Joel Jeffries, Alyssa Kruczek and Harper Niergarth |
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Vol. 17 (2024), No. 4, 651–668
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Abstract |
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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 . For the normalized adjacency, spectral faux trees exist when the number of vertices, , is at least 4, and we give an explicit construction for a family whose size grows exponentially with for , where is a fixed integer. |
Keywords
cospectral, trees, faux trees, signless Laplacian matrix,
normalized adjacency matrix, characteristic polynomials
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Mathematical Subject Classification
Primary: 05C50
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Milestones
Received: 13 January 2023
Revised: 23 May 2023
Accepted: 27 May 2023
Published: 2 October 2024
Communicated by Glenn Hurlbert
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Authors |
| Steve Butler | |
| Iowa State University Ames, IA United States |
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| Elena D’Avanzo | |
| Carleton College Northfield, MN United States |
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| Rachel Heikkinen | |
| Augustana College Rock Island, IL United States |
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| Joel Jeffries | |
| Iowa State University Ames, IA United States |
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| Alyssa Kruczek | |
| Susquehanna University Selinsgrove, PA United States |
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| Harper Niergarth | |
| University of Minnesota–Twin
Cities Minneapolis, MN United States |
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| © 2024 MSP (Mathematical Sciences Publishers). |
