Algebraic Graph Theory

A revision of an important textbook: essential reading for all combinatorialists.

This book presents and illustrates the main tools and ideas of algebraic graph theory, with a primary emphasis on current rather than classical topics. It is designed to offer self-contained treatment of the topic, with strong emphasis on concrete examples.

There is no other book with such a wide scope of both areas of algebraic graph theory.

This is a highly self-contained book about algebraic graph theory which is written with a view to keep the lively and unconventional atmosphere of a spoken text to communicate the enthusiasm the author feels about this subject. The focus is on homomorphisms and endomorphisms, matrices and eigenvalues. Graph models are extremely useful for almost all applications and applicators as they play an important role as structuring tools. They allow to model net structures - like roads, computers, telephones - instances of abstract data structures - like lists, stacks, trees - and functional or object oriented programming.

In this substantial revision of a much-quoted monograph first published in 1974, Dr. Biggs aims to express properties of graphs in algebraic terms, then to deduce theorems about them. In the first section, he tackles the applications of linear algebra and matrix theory to the study of graphs; algebraic constructions such as adjacency matrix and the incidence matrix and their applications are discussed in depth. There follows an extensive account of the theory of chromatic polynomials, a subject that has strong links with the "interaction models" studied in theoretical physics, and the theory of knots. The last part deals with symmetry and regularity properties. Here there are important connections with other branches of algebraic combinatorics and group theory. The structure of the volume is unchanged, but the text has been clarified and the notation brought into line with current practice. A large number of "Additional Results" are included at the end of each chapter, thereby covering most of the major advances in the past twenty years. This new and enlarged edition will be essential reading for a wide range of mathematicians, computer scientists and theoretical physicists.

In this substantial revision of a much-quoted monograph first published in 1974, Dr. Biggs aims to express properties of graphs in algebraic terms, then to deduce theorems about them. In the first section, he tackles the applications of linear algebra and matrix theory to the study of graphs; algebraic constructions such as adjacency matrix and the incidence matrix and their applications are discussed in depth. There follows an extensive account of the theory of chromatic polynomials, a subject that has strong links with the "interaction models" studied in theoretical physics, and the theory of knots. The last part deals with symmetry and regularity properties. Here there are important connections with other branches of algebraic combinatorics and group theory. The structure of the volume is unchanged, but the text has been clarified and the notation brought into line with current practice. A large number of "Additional Results" are included at the end of each chapter, thereby covering most of the major advances in the past twenty years. This new and enlarged edition will be essential reading for a wide range of mathematicians, computer scientists and theoretical physicists.

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 32. Chapters: Adjacency algebra, Adjacency matrix, Algebraic connectivity, Alpha centrality, Cayley graph, Clustering coefficient, Conductance (graph), Conference graph, Conference matrix, Cycle space, Degree matrix, Distance-regular graph, Distance-transitive graph, Dual graph, Edge-transitive graph, Edge space, Frucht's theorem, Graph automorphism, Graph energy, Half-transitive graph, Ihara zeta function, Incidence matrix, Integral graph, Kirchhoff's theorem, Laplacian matrix, Lovasz conjecture, Mac Lane's planarity criterion, Matching polynomial, Modularity (networks), Parry-Sullivan invariant, Ramanujan graph, Rank (graph theory), Seidel adjacency matrix, Semi-symmetric graph, Spectral graph theory, Strongly regular graph, Tutte matrix, Two-graph, Vertex-transitive graph.

Algebra and Graph Theory are two fascinating branches of Mathematics. The tools of each have been used in the other to explore and investigate problems in depth. Especially the Cayley graphs constructed out of the group structures have been greatly and extensively used in Parallel computers to provide network to the routing problem. ALGEBRA, GRAPH THEORY AND THEIR APPLICATIONS takes an inclusive view of the two areas and presents a wide range of topics. It includes sixteen referred research articles on algebra and graph theory of which three are expository in nature alongwith articles exhibiting the use of algebraic techniques in the study of graphs. A substantial proportion of the book covers topics that have not yet appeared in book form providing a useful resource to the younger generation of researchers in Discrete Mathematics.

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 31. Chapters: Centrality, Cayley graph, Clustering coefficient, Symmetric graph, Graph automorphism, Adjacency matrix, Modularity, Frucht's theorem, Kirchhoff's theorem, Incidence matrix, Distance-transitive graph, Conference matrix, Matching polynomial, Cycle space, Algebraic connectivity, Distance-regular graph, Strongly regular graph, Lov sz conjecture, Dual graph, Laplacian matrix, Spectral graph theory, Vertex-transitive graph, Ramanujan graph, Two-graph, Conductance, Semi-symmetric graph, Edge space, Ihara zeta function, Half-transitive graph, Edge-transitive graph, Seidel adjacency matrix, Adjacency algebra, Degree matrix, Parry-Sullivan invariant, Conference graph, Integral graph, Rank. Excerpt: Within graph theory and network analysis, there are various measures of the centrality of a vertex within a graph that determine the relative importance of a vertex within the graph (for example, how important a person is within a social network, or, in the theory of space syntax, how important a room is within a building or how well-used a road is within an urban network). There are four measures of centrality that are widely used in network analysis: degree centrality, betweenness, closeness, and eigenvector centrality. For a review as well as generalizations to weighted networks, see Opsahl et al. (2010). The

This book presents and illustrates the main tools and ideas of algebraic graph theory, with a primary emphasis on current rather than classical topics. It is designed to offer self-contained treatment of the topic, with strong emphasis on concrete examples.