This groundbreaking book is the first comprehensive resource dedicated to the Quadratic Embedding Constant (QEC), the novel numeric invariant that precisely bridges combinatorial graph theory and Euclidean distance geometry. Written by the creator of the QEC, the text presents a systematic and in-depth study of how graphs can be characterized and classified through this invariant, with particular emphasis on its behavior under graph operations such as Star Products and Graph Joins.Discover how the QEC arises directly from Schoenberg's foundational result — that a graph admits a Quadratic Embedding (QE) if its distance matrix is conditionally negative definite. Learn about the wider significance of QE, which is rooted in Menger's work and appears in harmonic analysis of discrete groups and quantum probability. Furthermore, gain deep insight into distance spectra (distance matrix eigenvalues) and the way the QEC interlaces the largest and second largest eigenvalues.This self-contained and accessible text is perfect for both non-experts and advanced researchers.