Algebraic Graph Theory Seminar - Ada Chan

Title: Complex Hadamard diagonalizable graphs

Abstract: 

A graph is complex Hadamard diagonalizable if its Laplacian matrix is diagonalizable by a complex Hadamard matrix.

This is a natural generalization of the Hadamard diagonalizable graphs introduced by Barik, Fallat and Kirkland.

My interest in these graphs is two-fold:

• Every pair of vertices in a complex Hadamard diagonalizable graph are cospectral, which is a necessary condition for perfect state transfer to occur in the continuous quantum walk on these graphs. These graphs are good candidates to have perfect state transfer in their continuous walks.
• Complex Hadamard matrices are examples of type-II matrices. Nomura's construction gives a formally dual pair of association scheme from a type-II matrix. Every graph in the Nomura algebra of a complex Hadamard matrix is complex Hadamard diagonalizable.

In this talk, we present some progress on the continuous quantum walks of complex Hadamard diagonalizable graphs.  

We discuss the Nomura algebra of type-II matrices and some related open problems.