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

REFERENCES

Cahit, I. "Homepage for the Graph Labelling Problems and New Results." http://www.emu.edu.tr/~cahit/CORDIAL.htm.

Gallian, J. "Dynamic Survey of Graph Labeling." Elec. J. Combin. DS6. Dec. 21, 2018. https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS6.

Gilbert, E. N. "Enumeration of Labeled Graphs." Canad. J. Math. 8, 405-411, 1956.

Harary, F. "Labeled Graphs." Graph Theory. Reading, MA: Addison-Wesley, pp. 10 and 178-180, 1994.

Harary, F. and Palmer, E. M. "Labeled Enumeration." Ch. 1 in Graphical Enumeration. New York: Academic Press, pp. 1-31, 1973.

Sloane, N. J. A. Sequences A001187/M3671, A006125/M1897, A095340 and A095351 in "The On-Line Encyclopedia of Integer Sequences.

"Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995.