ExtractCycles[g] gives a maximal list of edge-disjoint cycles in graph g.

MaximumSpanningTree[g] uses Kruskal's algorithm to find a maximum spanning tree of graph g.

NumberOfSpanningTrees[g] gives the number of labeled spanning trees of graph g.

PlanarQ[g] yields True if graph g is planar, meaning it can be drawn in the plane so no two edges cross.

PseudographQ[g] yields True if graph g is a pseudograph, meaning it contains self-loops.

Radius[g] gives the radius of graph g, the minimum eccentricity of any vertex of g.

SelfComplementaryQ[g] yields True if graph g is self-complementary, meaning it is isomorphic to its complement.

SelfLoopsQ[g] yields True if graph g has self-loops.

SimpleQ[g] yields True if g is a simple graph, meaning it has no multiple edges and contains no self-loops.

TransitiveReduction[g] finds a smallest graph that has the same transitive closure as g.