# Classifying Quantum Adjacency Matrices

## Faculty Mentor Name

Lara Ismert, Mitch Hamidi

## Format Preference

Poster

## Abstract

A finite directed graph consists of a finite set of vertices and an adjacency matrix that describes when there is an edge from one vertex to another. A quantum graph replaces the finite vertex set with a finite-dimensional matrix algebra and replaces the adjacency matrix with a quantum adjacency matrix, which is a function on the matrix algebra. In this talk, we classify certain classes of quantum adjacency matrices by computing their eigenvalues and determining when they are (quantum) regular or when they are homomorphisms. Additionally, we study relationships that arise between the eigenvalues of a quantum graph’s quantum adjacency matrix and its corresponding quantum edge checker, which is analogous to a graph’s edge matrix and is used to determine the quantum graph’s “edges.”

Classifying Quantum Adjacency Matrices

A finite directed graph consists of a finite set of vertices and an adjacency matrix that describes when there is an edge from one vertex to another. A quantum graph replaces the finite vertex set with a finite-dimensional matrix algebra and replaces the adjacency matrix with a quantum adjacency matrix, which is a function on the matrix algebra. In this talk, we classify certain classes of quantum adjacency matrices by computing their eigenvalues and determining when they are (quantum) regular or when they are homomorphisms. Additionally, we study relationships that arise between the eigenvalues of a quantum graph’s quantum adjacency matrix and its corresponding quantum edge checker, which is analogous to a graph’s edge matrix and is used to determine the quantum graph’s “edges.”