Research Synopsis

My chosen research topic for the Graduate Research Seminar is in the area of Spectral Graph Theory, which is the study of graphs from the perspective of the eigenvalues of their adjacency matrix. Specifically, I am interested in the class of graphs known as Integral Graphs. These are graphs who's spectrum consists of only integers.

Spectral Graph Theory is interesting to me since it combines graph theory, which I have been working on for the past year, and Linear Algebra, which is a subject that I have always enjoyed but never had the opportunity to apply to my research.

I have considered the Bruhat graphs that arise from the Weyl Groups associated with Root System of various Lie Types. I put the most emphasis on type A, where I computed exhaustively for a small number of generators. This lead to the observation that the full Bruhat graph is integral. Also, these structures are used to index Schubert varieties, the smooth Schubert varieties appear to be consistently associated with integral Bruhat graphs.

For specifics on what questions I am looking to answer, it is best to consult my most recent assignments from the course. (As keeping this page up to day may not be feasible.)

A few of my references I am studying are linked below (This list will grow or shrink as I consider my references.):

Constant Time Computation of Minimum Dominating Sets by Marilynn Livingston and Quentin F. Stout

Constructing fifteen infinite classes of nonregular bipartite integral graphs by Ligong Wang and Cornelis Hoede

A Survey on Integral Graphs by K. Balinska, D. Cvetkovic, Z. Radosavljevic, S. Simi´c, D. Stevanovic

A survey of results on integral trees and integral graphs by Ligong Wang

Eigen Spaces of Graphs by Dragos Cvetkovic, Peter Rowlinson, and Slobodan Simic

Spectra of Graphs Theory and Application by Dragos Cvetkovic, Michael Doob, and Horst Sachs