A complete tripartite graph is the 
case of a complete
k-partite graph. In other words, it is a tripartite graph (i.e., a set of graph vertices decomposed
into three disjoint sets such that no two graph
vertices within the same set are adjacent)
such that every vertex of each set
of graph vertices is adjacent
to every vertex in the other two sets.
If there are 
,
,
and 
graph vertices in the three sets,
the complete tripartite graph (sometimes also called a complete trigraph) is denoted
.
Special classes are summarized in the following table.
 | special class |
 |  -fan graph |
 |  -fan graph |
 |  -cone graph |
Some special cases are summarized in the following table, some of which are illustrated above.
 | name |  |
| 3 | triangle graph |  |
| 4 | diamond graph |  |
| 5 | (3,2)-fan graph |  |
| 5 | 5-wheel graph |  |
| 6 | octahedral graph  |  |
| 6 | (3,3)-fan graph |  |
| 6 | (4,2)-fan graph |  |
| 7 | (4,3)-cone graph |  |
| 7 | (4,3)-fan graph |  |
| 7 | (5,2)-fan graph |  |
| 8 | (4,4)-cone graph |  |
| 8 | (5,3)-fan graph |  |
| 9 | 9-circulant
graph  |  |
| 9 | (4,5)-cone graph |  |
| 12 | circulant graph  |  |
| 15 | circulant graph  |  |
The number of Hamiltonian cycles in the graph 
can be efficiently computed using a general complete
k-partite graph recurrence (Horák
and Tovarek 1979). A closed-form formula is also known (submitted as a problem, 2025).
Spectral comparisons ask which nonisomorphic graph on 
vertices is closest to 
in graph spectral
distance. Jovanović et al. (2026) refuted two conjectures proposing
as this minimizer for 
. Their counterexample 
is obtained from 
by adjoining a new graph
vertex adjacent to every graph vertex in the
part of size 
and to no others. For the taxicab metric 
and Euclidean metric 
on the ordered spectra, 
and 
. The second limit is for the squared
distance, not the distance itself.
-Partite Graphs." Math. Slovaca 29, 43-47,
1979.House of Graphs. Complete Tripartite Graphs.