By providing a completely extensible set of vertex and edge properties, you can make graphs represent much more than the structural information embodied in their topology. For instance, vertices could contain dynamic system models and edges could contain signals, and the graph could then represent a block-diagram model. The vertex and edge properties can also be used to store the state when scanning the graph in a depth-first or breadth-first manner as used by many graph algorithms. But the Wolfram Language provides a multi-paradigm approach to graph programming with a rich set of matrix, optimization, and Boolean-based frameworks that allow for a variety of extensions to the whole graph framework.
BreadthFirstScan — scan a graph in a breadth-first order
DepthFirstScan — scan a graph in a depth-first order
AdjacencyMatrix — convert to a sparse adjacency matrix representation
AdjacencyGraph — construct a graph from an adjacency matrix
Minimize — minimize a function with constraints