Experimental Mathematics

Survey of spectra of Laplacians on finite symmetric spaces

Audrey Terras

Abstract

We survey what is known about spectra of combinatorial Laplacians (or adjacency operators) of graphs on the simplest finite symmetric spaces. This work is joint with J. Angel, N. Celniker, A. Medrano, P. Myers, S. Poulos, H. Stark, C. Trimble, and E. Velasquez. For each finite field $\Fq$ with $q$ odd, we consider graphs associated to finite Euclidean and non-Euclidean symmetric spaces over $\Fq$. We are mainly interested in three questions regarding the eigenvalues and eigenfunctions of the combinatorial Laplacian as $q$ goes to infinity: How large is the second largest eigenvalue, in absolute value, compared with the graph's degree? (The largest eigenvalue is the degree.) What can one say about the distribution of eigenvalues? What can one say about the "level curves'' of the eigenfunctions?

Article information

Source
Experiment. Math., Volume 5, Issue 1 (1996), 15-32.

Dates
First available in Project Euclid: 13 March 2003

Permanent link to this document
https://projecteuclid.org/euclid.em/1047591144

Mathematical Reviews number (MathSciNet)
MR1412951

Zentralblatt MATH identifier
0871.05044

Subjects
Primary: 11T99: None of the above, but in this section
Secondary: 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.) 11T23: Exponential sums 20H30: Other matrix groups over finite fields

Citation

Terras, Audrey. Survey of spectra of Laplacians on finite symmetric spaces. Experiment. Math. 5 (1996), no. 1, 15--32. https://projecteuclid.org/euclid.em/1047591144


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