Turán graph

Turán graph
The Turán graph T(13,4)
Named after	Pál Turán
Vertices	n
Edges	~${\displaystyle {\frac {(r-1)n^{2}}{2r}}}$
Radius	${\displaystyle \left\{{\begin{array}{ll}\infty &r=1\\2&r\leq n/2\\1&{\text{otherwise}}\end{array}}\right.}$
Diameter	${\displaystyle \left\{{\begin{array}{ll}\infty &r=1\\1&r=n\\2&{\text{otherwise}}\end{array}}\right.}$
Girth	${\displaystyle \left\{{\begin{array}{ll}\infty &r=1\vee (n\leq 3\wedge r\leq 2)\\4&r=2\\3&{\text{otherwise}}\end{array}}\right.}$
Chromatic number	r
Notation	T(n,r)

The Turán graph T(n,r) is a complete multipartite graph formed by partitioning a set of n vertices into r subsets, with sizes as equal as possible, and connecting two vertices by an edge if and only if they belong to different subsets. The graph will have ${\displaystyle (n\,{\bmod {\,}}r)}$ subsets of size ${\displaystyle \lceil n/r\rceil }$, and ${\displaystyle r-(n\,{\bmod {\,}}r)}$ subsets of size ${\displaystyle \lfloor n/r\rfloor }$. That is, it is a complete r-partite graph