Extensions and generalizations of graphs

assignment of colors to elements of a graph subject to certain constraints

graph with oriented edges

frame|An example of an undirected hypergraph, with X = \{v_1, v_2, v_3, v_4, v_5, v_6, v_7\} and E = \{e_1,e_2,e_3,e_4\} = \{\{v_1, v_2, v_3\}, \{v_2,v_3\}, \{v_3,v_5,v_6\}, \{v_4\}\}. This hypergraph has order 7 and size 4. Here, edges do not just connect two vertices but several, and are represented by colors. alt=PAOH visualization of a hypergraph|thumb|Alternative representation of the hypergraph reported in the figure above, called PAOH. Edges are vertical lines connecting vertices. V7 is an isolated vertex. Vertices are aligned to the left. The legend on the right shows the names of the

thumb|right|A multigraph with multiple edges (red) and several loops (blue). Not all authors allow multigraphs to have loops. In mathematics, and more specifically in graph theory, a multigraph is a graph which is permitted to have multiple edges (also called parallel edges), that is, edges that have the same end nodes. Thus two vertices may be connected by more than one edge.

assignment of labels, traditionally represented by integers, to the edges or vertices, or both, of a graph

graph that is permitted to contain both directed and undirected edges

metric graph equipped with a differential operator acting on functions on the graph

undirected graph in which one vertex has been distinguished as the root