Mathematics Lecture Series, Fall 2012

Fresno State

Dr. Stefaan Delcroix

California State University, Fresno

“Fractals and Graphs – a beautiful connection”

Friday, November 2, 2012

from 4:00 p.m. to 5:00 p.m.

Science 145

We start with some basic definitions in graph theory: incidence matrix, eigenvalues and cartesian product. Then we introduce the tensor product of vector spaces. As the main result, we use tensor products to prove a relation between the eigenvalues of the cartesian product of graphs and the eigenvalues of the original graphs. Next, we introduce the concept of the Hausdorff dimension of a fractal and the relation with the eigenvalues of a strongly connected, weighted, directed graph. Finally, we use the Perron-Frobenius Theorem to calculate the Haussdorf dimension of fractals and cartesian products of graphs. We give explicit formulas in terms of the eigenvalues of the graph and the similarity ratios used with each graph.

For further information, please contact Dr. Tamas Forgacs at (559) 278-4907 or e-mail tforgacs@csufresno.edu.