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 ![V_n=<span style=]()
{1,2,...,n}" src="https://mathworld.wolfram.com/images/equations/LabeledGraph/Inline5.svg" style="height:22px; width:137px" />, the number of vertex-labeled graphs is given by 
. Two graphs 
and 
with graph vertices ![V_n=<span style=]()
{1,2,...,n}" src="https://mathworld.wolfram.com/images/equations/LabeledGraph/Inline9.svg" style="height:22px; width:137px" /> are said to be isomorphic if there is a permutation 
of 
such that ![<span style=]()
{u,v}" src="https://mathworld.wolfram.com/images/equations/LabeledGraph/Inline12.svg" style="height:22px; width:41px" /> is in the set of graph edges 
iff ![<span style=]()
{p(u),p(v)}" src="https://mathworld.wolfram.com/images/equations/LabeledGraph/Inline14.svg" style="height:22px; width:97px" /> 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.
