Math 9412, Spectral Graph Theory Summer 2015

Instructor: Masoud Khalkhali Mathematics Department, University of Western Ontario London, ON, Canada

• Course Outline: Spectral graph theory mostly deals with spectrum of the Laplacian of a graph. While graph Laplacians have many similarities with Laplacian of a Rieman- nian manifold, the study of them requires less technical know how. In par- ticular the only prerequisite for this course is a good understanding of un- dergraduate linear algebra. Also no prior familiarity with graph theory will be assumed. The advantage of working with graphs is that many ideas of spectral geometry and analytic numbers theory can be studied in a very con- crete manner. With the ever increasing role played by large networks and big data in every day life, ideas developed in spectral graph theory are gaining importance in many fields now.

• Time permitting, we shall try to cover some of the following topics: Laplacian of a graph and its eigenvalues, Isospectral but not isometric graphs, Random walks on graphs, Markov chains, Ergodicity, Dirichlet boundary value problems for graphs, Phase transition in planar Ising model, Matrix-tree theorem, Heat kernel techniques, Counting periodic orbits, Ihara zeta function of a graph, and its Riemann hypothesis.

• Textbook: Selected sections from

1 1) Spectral Graph Theory by F. R. K. Chung (available online), 2) Zeta functions of graphs by A. Terras (online version available through Taylor library).

• Marking scheme: Class participation and interaction will be essential. There will be 4 sets of assignments on topics covered in the course. Students are expected to choose a topic in consultation with me, prepare a short essay and present it at the end of classes.

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