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This paper describes applications of graph theory to some of the problems of network analysis and synthesis, starting with the early days of network theory. The first section is devoted to the classical results related to the topological analysis of linear, passive, and transfornerless networks. Then it goes over to contributions dealing with generalization of these methods to networks with mutual couplings and active elements. Appearance of integrated circuits and digital computers provided both motivation and means for studying large-scale systems. A part of the paper is devoted to publications on large-scale network analysis dealing with diacoptics and graph theoretic methods for defining efficient Gaussian elimination algorithms for solution of sparse sets of linear algebraic equations. The part on topological synthesis starts with the early results dealing with relations between the structure and the properties of passive, reciprocal networks without mutual inductances. The last section is devoted to contributions in the field of topological synthesis of pure resistive networks dealing with such topics as conditions of realizability, minimal realizations and bounds on the number of port-terminals.