This book studies algebraic representations of graphs in order to investigate combinatorial structures via local symmetries. Topological, combinatorial and algebraic classifications are distinguished by invariants of polynomial type and algorithms are designed to determine all such classifications with complexity analysis. Being a summary of the author's original work on graph embeddings, this book is an essential reference for researchers in graph theory. ContentsAbstract GraphsAbstract MapsDualityOrientabilityOrientable MapsNonorientable MapsIsomorphisms of MapsAsymmetrizationAsymmetrized Petal BundlesAsymmetrized MapsMaps within SymmetryGenus PolynomialsCensus with PartitionsEquations with PartitionsUpper Maps of a GraphGenera of a GraphIsogemial GraphsSurface Embeddability | Algebraic Elements of Graphs by Yanpei Liu, Hardcover | Indigo Chapters