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Spectra of infinite graphs via freeness with amalgamation

Part of: Graph theory Special classes of linear operators Selfadjoint operator algebras

Published online by Cambridge University Press:  03 October 2022

Jorge Garza-Vargas Open the ORCID record for Jorge Garza-Vargas [Opens in a new window]  and
Archit Kulkarni Open the ORCID record for Archit Kulkarni [Opens in a new window]
Jorge Garza-Vargas*
Affiliation:
Department of Mathematics, University of California, Berkeley, Berkeley, CA, USA e-mail: akulkarni@berkeley.edu
Archit Kulkarni
Affiliation:
Department of Mathematics, University of California, Berkeley, Berkeley, CA, USA e-mail: akulkarni@berkeley.edu
*
e-mail: jgarzavargas@berkeley.edu
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Abstract

We use tools from free probability to study the spectra of Hermitian operators on infinite graphs. Special attention is devoted to universal covering trees of finite graphs. For operators on these graphs, we derive a new variational formula for the spectral radius and provide new proofs of results due to Sunada and Aomoto using free probability.

With the goal of extending the applicability of free probability techniques beyond universal covering trees, we introduce a new combinatorial product operation on graphs and show that, in the noncommutative probability context, it corresponds to the notion of freeness with amalgamation. We show that Cayley graphs of amalgamated free products of groups, as well as universal covering trees, can be constructed using our graph product.

Keywords

Free probabilityinfinite graphsspectral theoryJacobi matrices

MSC classification

Primary: 46L54: Free probability and free operator algebras
Secondary: 05C63: Infinite graphs 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.) 47B36: Jacobi (tridiagonal) operators (matrices) and generalizations
Type
Article
Information
Canadian Journal of Mathematics , Volume 75 , Issue 5 , October 2023 , pp. 1633 - 1684
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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