Isospectral reductions and quantum walks on graphs

TL;DR

This paper introduces a new formula for isospectral reductions of matrices and graphs, generalizes the concept, and explores their application to quantum walks, leading to the construction of graphs with perfect quantum state transfer.

Contribution

It provides a novel formula for isospectral reductions, generalizes the concept, and links reductions to quantum walk transition matrices, enabling new graph constructions.

Findings

New formula for isospectral reduction of matrices and graphs

Generalization of isospectral reductions

Construction of graphs with perfect quantum state transfer

Abstract

We give a new formula for computing the isospectral reduction of a matrix (and graph) down to a submatrix (or subgraph). Using this, we generalize the notion of isospectral reductions. In addition, we give a procedure for constructing a matrix whose isospectral reduction down to a submatrix is given. We also prove that the isospectral reduction completely determines the restriction of the quantum walk transition matrix to a subset. Using these, we construct new families of simple graphs exhibiting perfect quantum state transfer.