Labeled Graph

A labeled graph is a finite series of graph vertices with a set of graph edges of 2-subsets of . Given a graph vertex set , the number of vertex-labeled graphs is given by . Two graphs and with graph vertices are said to be isomorphic if there is a permutation of such that is in the set of graph edges iff is in the set of graph edges .

The term "labeled graph" when used without qualification means a graph with each node labeled differently (but arbitrarily), so that all nodes are considered distinct for purposes of enumeration. The total number of (not necessarily connected) labeled -node graphs for , 2, ... is given by 1, 2, 8, 64, 1024, 32768, ... (OEIS A006125; illustrated above), and the numbers of connected labeled graphs on -nodes are given by the logarithmic transform of the preceding sequence, 1, 1, 4, 38, 728, 26704, ... (OEIS A001187; Sloane and Plouffe 1995, p. 19).

The numbers of graph vertices in all labeled graphs of orders , 2, ... are 1, 4, 24, 256, 5120, 196608, ... (OEIS A095340), which the numbers of edges are 0, 1, 12, 192, 5120, 245760, ... (OEIS A095351), the latter of which has closed-form

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