VertexList[g] gives a list of all vertices in the graph g.

VertexDegree[g] gives the list of vertex degrees for all vertices in the graph g.VertexDegree[g, v] gives the vertex degree for the vertex v.

KCoreComponents[g, k] gives the k-core components of the underlying simple graph of g.KCoreComponents[g, k, "InDegree"] gives the k-core components with vertex in-degrees at ...

GraphDiameter[g] gives the maximum eccentricity of the vertices in the graph g.

VertexShape is an option and property for Graph and related functions that specifies the graphics used for vertices.

GraphDifference[g, h] constructs the graph resulting from subtracting the edges of graph h from the edges of graph g.

Hypercube[n] constructs an n-dimensional hypercube.

VertexCoverQ[g, vlist] yields True if the vertex list vlist is a vertex cover of the graph g, and False otherwise.

BipartiteMatchingAndCover[g] takes a bipartite graph g and returns a matching with maximum weight along with the dual vertex cover. If the graph is not weighted, it is ...

BipartiteMatching[g] gives the list of edges associated with a maximum matching in bipartite graph g. If the graph is edge weighted, then the function returns a matching with ...