Irregularity in Graphs

In 1891, the Danish mathematician Julius Petersen published Die Theorie der regularen Graphen, known in English as The Theory of Regular Graphs. It is generally regarded as the earliest paper on graphs written in a purely theoretical spirit. The study of regular graphs, in which every vertex has the same degree, has remained a widely pursued and popular subject in the 130 years since.

The term regular can also apply more broadly to any graph that shares some common feature across its whole structure. Over the last several decades, mathematicians have shown growing interest in the reverse idea: graphs that are, in a sense, the opposite of regular. This book explores that direction and develops the notion of irregularity in graphs.

The book surveys a range of irregularity concepts across different areas of graph theory. Topics include vertex degrees, graph labelings, weightings, colorings, graph structures, Eulerian and Hamiltonian properties, graph decompositions, and Ramsey-type problems.