Algebraic Graph Theory Seminar - Xiaohong Zhang

Title: Perfect state transfer on Hadamard diagonalizable graphs

Abstract:

A (weighted) graph whose Laplacian matrix is diagonalizable by a Hadamard matrix is said to be Hadamard diagonalizable. Such a graph is regular, therefore it exhibits the same diagonalization property and state transfer property no matter whether the Laplacian or adjacency matrix is used. Several families of graphs are known to be Hadamard diagonalizable, for example, the complete graph on 4n vertices, the balanced complete bipartite graph on 4n vertices (provided that there is a Hadamard matrix of that order), and cubelike graphs.

Perfect state transfer properties of Hadamard diagonalizable graphs are well studied. I will show a connection between cubelike graphs and Sylvester Hadamard matrices, an eigenvalue characterization of when an integer-weighted graph that is diagonalized by a given Hadamard matrix admits PST at time $\pi/2$. We also introduce a weighted graph operation--the merge, under which Hadamard diagonalizability is kept. A complete characterization when the merge of two integer-weighted graphs that are diagonalizable by the same normalized Hadamard matrix admits PST is given.