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
See also
15 Puzzle, A-Cordial Graph, Connected Graph, Cordial Graph, Edge-Graceful Graph, Elegant Graph, Graceful Graph, Graph, h-Cordial Graph, Harmonious Graph, Labeled Digraph, Labeled Tree, Magic Graph, Oriented Graph, Taylor's Condition, Unlabeled Graph, Weighted Tree
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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. Oct. 30, 2025. 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.
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Cite this as:
Weisstein, Eric W. "Labeled Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LabeledGraph.html