Isospectral Graph Reductions, Estimates of Matrices' Spectra, and Eventually Negative Schwarzian Systems

Dissertation Defense
Tuesday, March 8, 2011 - 09:00
1 hour (actually 50 minutes)
Skiles 006
School of Mathematics, Georgia Tech
Real world networks typically consist of a large number of dynamical units with a complicated structure of interactions. Until recently such networks were most often studied independently as either graphs or as coupled dynamical systems. To integrate these two approaches we introduce the concept of an isospectral graph transformation which allows one to modify the network at the level of a graph while maintaining the eigenvalues of its adjacency matrix. This theory can then be used to rewire dynamical networks, considered as dynamical systems, in order to gain improved estimates for whether the network has a unique global attractor. Moreover, this theory leads to improved eigenvalue estimates of Gershgorin-type. Lastly, we will discuss the use of Schwarzian derivatives in the theory of 1-d dynamical systems.