Applications of combinatorial matrix theory to Laplacian matrices of graphs Preface On the surface matrix theory and graph theory are seemingly very different branches of mathematics. However these two branches of mathematics interact since it is often convenient to represent a graph as a matrix. Adjacency Laplacian and incidence matrices are commonly used to represent graphs. In 1973 Fiedler published his first paper on Laplacian matrices of graphs and showed how many properties of the Laplacian matrix especially the eigenvalues can give us useful information about the structure of the graph. Since then many papers have been published on Laplacian matrices. This book is a compilation of many of the exciting results concerning Laplacian matrices that have been developed since the mid 1970s. Papers written by well known mathematicians such as (alphabetically) Fallat Fiedler Grone Kirkland Merris Mohar Neumann Shader Sunder and several others are consolidated here. Each theorem is referenced to its appropriate paper so that the reader can easily do more in depth research on any topic of interest. However the style of presentation in this book is not meant to be that of a journal but rather a reference textbook. Therefore more examples and more detailed calculations are presented in this book than would be in a journal article. Additionally most sections are followed by exercises to aid the reader in gaining a deeper understanding of the material. Some exercises are routine calculations that involve applying the theorems presented in the section. Other exercises require a more in depth analysis of the theorems and require the reader to prove theorems that go beyond what was presented in the section. Many of these exercises are taken from relevant papers and they are referenced accordingly