Editing Turán graph (section)

==Other properties==
Every Turán graph is a [[cograph]]; that is, it can be formed from individual vertices by a sequence of [[disjoint union]] and [[complement (graph theory)|complement]] operations. Specifically, such a sequence can begin by forming each of the independent sets of the Turán graph as a disjoint union of isolated vertices. Then, the overall graph is the complement of the disjoint union of the complements of these independent sets.

{{harvtxt|Chao|Novacky|1982}} show that the Turán graphs are ''chromatically unique'': no other graphs have the same [[chromatic polynomial]]s. Nikiforov (2005) uses Turán graphs to supply a lower bound for the sum of the ''k''th [[eigenvalue]]s of a graph and its complement.{{sfnp|Chao|Novacky|1982}}

{{harvtxt|Falls|Powell|Snoeyink|2003}} develop an efficient algorithm for finding clusters of orthologous groups of genes in genome data, by representing the data as a graph and searching for large Turán subgraphs.{{sfnp|Falls|Powell|Snoeyink|2003}}

Turán graphs also have some interesting properties related to [[geometric graph theory]]. {{harvtxt|Pór|Wood|2005}} give a lower bound of Ω((''rn'')<sup>3/4</sup>) on the volume of any three-dimensional [[Graph drawing|grid embedding]] of the Turán graph.{{sfnp|Pór|Wood|2005}} {{harvtxt|Witsenhausen|1974}} conjectures that the maximum sum of squared distances, among ''n'' points with unit diameter in '''R'''<sup>''d''</sup>, is attained for a configuration formed by embedding a Turán graph onto the vertices of a regular simplex.{{sfnp|Witsenhausen|1974}}

An ''n''-vertex graph ''G'' is a [[Glossary of graph theory#Subgraphs|subgraph]] of a Turán graph ''T''(''n'',''r'') if and only if ''G'' admits an [[equitable coloring]] with ''r'' colors. The partition of the Turán graph into independent sets corresponds to the partition of ''G'' into color classes. In particular, the Turán graph is the unique maximal ''n''-vertex graph with an ''r''-color equitable coloring.