A complete multipartite graph is a graph that is a complete k-partite graph for some positive integer 
(Chartrand and Zhang 2008, p. 41).
The term therefore denotes the class obtained by allowing the number 
of partite sets in a complete
k-partite graph to vary. If the partite sets have sizes 
, 
, ..., 
, the graph is denoted 
.
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 
.
