A balanced complete multipartite graph is a complete multipartite graph whose partite sets all have the same cardinality. Equivalently,
it is a complete multipartite graph of the form 
, also denoted 
.
The graph 
has vertex count 
and edge count 
. It is 
-regular, has clique
number and chromatic number equal to 
,
and has independence number 
. Its graph complement
is the disjoint union 
. For 
and 
, its graph diameter
is 2.
When 
, 
is the complete
graph on 
vertices. Other special cases include balanced complete
bipartite graphs 
for 
and balanced complete
tripartite graphs 
for 
.
For 
and 
,
every vertex of 
has vertex transmission
 |
so balanced complete multipartite graphs are transmission-regular.
