Background on the Physics on Graphs project
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Spectral Methods on Graphs
This project is directed by Dr. Stratos Prassidis.
One focus of the project has been the implementing of heat kernel methods to reprove and improve estimates of the spectral radius of infinite graphs. For infinite graphs the analogue of the adjacency matrix is an operator on the Hilbert space of functions on the vertex set of a graph. The spectrum of this operator is known to reflect properties of graphs. Theorems due to Kesten and Grigorchuk provide estimates for the spectral radius of Cayley graphs of groups, and other graphs G where the automorphism group of G acts transitively on G.
Another focus is on generalization of the zig-zag product of graphs and its spectral properties. Zig-zag products are a tool for constructing infinite sequences of expanders combinatorially. Zig-zag products of graphs correspond to Cayley graphs of semidirect product of groups. Thus, zig-zag products are the combinatorial analogues of semidirect products of groups. We investigate methods for constructing zig-zag products categorically, and using zig-zag products to produce infinite families of non-regular Ramanujan graphs.
Previous topics include:
