Mathematics and Statistics Seminar – Hermie Monterde
- Seminar will be held via Zoom
- Zoom ID: 995 7773 8891
- Passcode: M410
About the event
Seminar in Theory and Applications
of
Discrete Math, Linear Algebra and Number Theory
Department of Mathematics and Statistics
Zoom
October 14, Monday, 4:10 – 5:00 PM
Title: Weakly Hadamard diagonalizable graphs
Speaker: Hermie Monterde
Abstract: A matrix 𝑀 is said to be 𝑆-diagonalizable if there exists a matrix 𝑃 having entries from 𝑆 such that 𝑀=𝑃Δ𝑃^−1 for some diagonal matrix Δ. A graph 𝑋 is said to be 𝑆-diagonalizable if its Laplacian matrix 𝐿:=𝐷−𝐴 is 𝑆-diagonalizable (Johnston and Plosker, 2023). Here, 𝐷 is the diagonal matrix of vertex degrees of 𝑋 and 𝐴 is the adjacency matrix of 𝑋. In this talk, we survey results on 𝑆-diagonalizable graphs, where 𝑆={−1,0,1} or 𝑆={−1,1}. In particular, we will focus on the case where 𝑃 is a Hadamard (that is, 𝑃 has entries from {−1,1} and the columns of 𝑃 are pairwise orthogonal) or a weak Hadamard matrix (that is, 𝑃 has entries from {−1,0,1} and the nonconsecutive columns of 𝑃 are pairwise orthogonal). Our main motivation for considering these graphs is its potential application to theory of continuous-time quantum walks, where knowledge about eigenvectors of certain matrices associated to graphs (more specifically, their orthogonal projection matrices) is important in determining which types of quantum state transfer occur in a graph. Some of the material presented in this talk is joint work with Darian McLaren and Sarah Plosker.