Matrix representations of graphs go back a long time and are still in some areas the only way to represent graphs. Adjacency matrices represent adjacent vertices and incidence matrix vertex-edge incidences. Both are fully capable of representing undirected and directed graphs. Matrix representations provide a bridge to linear algebra-based algorithms for graph computation.
AdjacencyGraph — graph from an adjacency matrix
IncidenceGraph — graph from an incidence matrix
AdjacencyMatrix — vertex-vertex adjacency matrix
IncidenceMatrix — vertex-edge incidence matrix
SparseArray — create and represent a sparse matrix