# On Mahler Measures and Digraphs

Joshua Coyston

Research output: ThesisDoctoral Thesis

## Abstract

This thesis is based in the intersection of number theory and graph theory, introducing a link between the Mahler measure of a polynomial and specific types of graphs, which we refer to as digraphs. One of the main aims of this thesis is to find “small” Mahler measure values “from” digraphs. Our other aims are to further extend our knowledge of the digraphs presented, the theory linking Mahler measures and graphs more broadly and, finally, understanding when polynomials may share the same Mahler measure.

As such, this thesis has two distinctive themes which are intertwined together. In
parts of the thesis, primarily Chapters 1, 3 and 6, our focus is more theoretical, and we aim to introduce and survey existing ideas, both from the topics of Mahler measures and digraph theory, as well as build upon these and present new concepts. Other parts, particularly Chapters 2 and 5, are focused on calculating Mahler measure values from a practical viewpoint, including introducing a new method for calculating the Mahler measures of two-variable polynomials, as well as detailing how exactly we performed our experiments.

Meanwhile, Chapter 4 is a blend of these themes: whilst it focuses on developing our theoretical knowledge, it is done in a way to prepare us for our experiments. We round off the thesis in Chapter 7 by summarizing potential future work.

In most circumstances, we present Mahler measure values to 8 decimal places, though this convention is broken on occasions where appropriate.
Original language English Ph.D. Royal Holloway, University of London McKee, James, Supervisor 1 Jun 2022 Unpublished - 2022

## Keywords

• Mahler measure
• Mahler's measure
• Digraph
• Number Theory
• Graph Theory
• Royal Holloway
• Joshua Coyston
• Josh Coyston
• Coyston
• Kronecker-cyclotomic
• Cyclotomic graph
• Cyclotomic digraph
• Graph
• Equivalent digraphs
• Pendant paths
• Charged pendant paths
• Bridged digraphs
• Small Mahler Measures
• Small limit points
• Same Mahler Measure
• Equivalent Mahler Measures