Date of Award

August 2023

Degree Type


Degree Name

Master of Science



First Advisor

Jeb F Willenbring

Committee Members

Pamela E Harris, Kevin B McLeod


doubly stochastic, enumeration, hollow symmetric matrix, spectra, symmetric matrix


This work concerns the spectra of doubly stochastic matrices whose entries are rational numbers with a bounded denominator. When the bound is fixed, we consider the enumeration of these matrices and also the enumeration of the orbits under the action of the symmetric group.

In the case where the bound is two, we investigate the symmetric case. Such matrices are in fact doubly stochastic, and have a nice characterization when we are in the special case where the diagonal is zero. As a central tool to this investigation, we utilize Birkhoff's theorem that asserts that the doubly stochastic matrices are exactly the polytope defined by the convex hull of permutation matrices. In particular, we consider the spectra along segments in the Birkhoff polytope.

Included in

Mathematics Commons