Sage is open source software for algebra and geometry experimentation. It has a library of graph algorithms for graph generation, shortest paths, coloring, centrality, asteroidal triples, comparability and permutation graphs, weakly chordal graphs, and hyperbolicity, among others.
GraphTea is a WYSIWYG for graph generation and visualization. It allows for creating a graph by point-click.
yEd is a WYSIWYG for graph generation and visualization. It allows for creating a graph by point-click, and supports a wide range of customization (fonts, colors, etc).
NetworkX is a Python library for the creation, manipulation, and study of the structure, dynamics, and functions of complex networks. It has a library of a variety of graph algorithms and supports reading/writing many different graph formats.
NetworKit is a Python library toolkit for large-scale network analysis, in the size range from thousands to billions of edges. It implements graph algorithms with a focus on parallelism and scalability.
tikZ/PGF is a TeX package for generating graphics. It can create high-quality, high-precision graphs with a number of defined TeX commands that draw graphics.