This is a substantial revision of a much-quoted monograph, first published in 1974. The structure is unchanged, but the text has been clarified and the notation brought into line with current practice. Many 'Additional Results' are included at the end of each chapter, thereby covering most of the significant advances in the last twenty years. Professor Biggs' primary aim remains to express the properties of graphs in algebraic terms and then deduce theorems about them. In the first part, he tackles linear algebra and matrix theory applications 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 idea of knots. The last part deals with symmetry and regularity properties. Here there are meaningful connections with other branches of algebraic combinatorics and group theory. This new and enlarged edition will be essential reading for a wide range of mathematicians, computer scientists and theoretical physicists.