Magic squares, their origins lost in antiquity, are among the more popular mathematical recreations. Over the years a number of generalizations have been proposed, going back in the last century to Sedlacek (early 1960s) who asked whether "magic" ideas could be applied to graphs. Around the same time Kotzig and Rosa formulated the study of graph labelings, or valuations as they were first called.

In the last decade, there has been a resurgence of interest in "magic labelings" and other graph valuations, e.g., graceful labelings, due to a number of interesting results that have applications and are related to the problems of decomposing graphs into trees. Trees remain an elusive subject. From the pure mathematics viewpoint, no progress has been made in answering the question: Does every tree have an edge-magic total labeling? However, the corresponding problem for vertex-magic total labelings has been solved, and the details are examined in this volume. The book also contains a number of recent constructions of magic graphs and verifications that families of graphs are magic.

Key features:

• short historical account of the subject
• concise, self-contained text; systematic exposition from basic topics in graph theory to current research
• applications of theorems from graph theory and interesting counting arguments
• exercises, research problems covering a range of difficulties, extensive bibliography, index
• solutions to many exercises

This concise, self-contained exposition is unique in its focus on the theory of magic graphs/labeling and its applications to a number of new areas. It may serve as a graduate text for a special topics seminar in mathematics or computer science, or as a professional text for the researcher.