Apportionable matrices and gracefully labelled graphs

TL;DR

This paper introduces the concept of apportionment of matrices via similarity transformations, explores its connections to graph labelings and quantum walks, and establishes conditions for apportionability, including constructions of non-apportionable matrices.

Contribution

It initiates the study of matrix apportionment, links it to graph theory and quantum applications, and provides new theoretical results and constructions.

Findings

Every rank one matrix can be apportioned by a unitary similarity.

Some 2x2 matrices cannot be apportioned, showing limitations.

A necessary condition for unitary apportionment is established.

Abstract

To apportion a complex matrix means to apply a similarity so that all entries of the resulting matrix have the same magnitude. We initiate the study of apportionment, both by unitary matrix similarity and general matrix similarity. There are connections between apportionment and classical graph decomposition problems, including graceful labelings of graphs, Hadamard matrices, and equiangluar lines, and potential applications to instantaneous uniform mixing in quantum walks. The connection between apportionment and graceful labelings allows the construction of apportionable matrices from trees. A generalization of the well-known Eigenvalue Interlacing Inequalities using graceful labelings is also presented. It is shown that every rank one matrix can be apportioned by a unitary similarity, but there are $2\x 2$ matrices that cannot be apportioned. A necessary condition for a matrix to be apportioned by unitary matrix is established. This condition is used to construct a set of matrices with nonzero Lebesgue measure that are not apportionable by a unitary matrix.