This textbook explores harmonic morphisms between graphs—maps from one graph to another that preserve harmonic functions—and in what ways they can relate one graph to another. It demonstrates how such a morphism-theoretic approach, in which the graphs are treated as “variables” and the specific maps are the focus of analysis, can be beneficial in several modern application areas. 

To prepare readers for studying harmonic morphisms, the book begins with an overview of graph morphisms and their definitions and basic properties, followed by background on harmonic functions on graphs. Harmonic morphisms are then considered in detail and characterized using matrix-theoretic techniques to aid in proving results. Applications to counting problems, graph quotients, product graphs, and other graph constructions are then explored. Exercise sets and open problems are included throughout the text to provide additional direction and motivation.

Designed for advanced undergraduate and beginning graduate students in mathematics, the book is suitable for both classroom use and self-study. Readers should be familiar with the fundamentals of linear algebra, graph theory, and group theory. 

Caroline Melles has been teaching college mathematics in Maryland for decades. She lives in the Annapolis area with her family.

W. David Joyner retired from college teaching in 2018 to focus on writing. He lives on the Maryland Eastern Shore with his wife and two dogs.