, is a complete multipartite graph; it is formed by partitioning a set of

, this edge count can be more succinctly stated as

By the pigeonhole principle, every set of r + 1 vertices in the Turán graph includes two vertices in the same partition subset; therefore, the Turán graph does not contain a clique of size r + 1.

According to Turán's theorem, the Turán graph has the maximum possible number of edges among all (r + 1)-clique-free graphs with n vertices.

Keevash & Sudakov (2003) show that the Turán graph is also the only (r + 1)-clique-free graph of order n in which every subset of αn vertices spans at least

edges, if α is sufficiently close to 1.

[1] The Erdős–Stone theorem extends Turán's theorem by bounding the number of edges in a graph that does not have a fixed Turán graph as a subgraph.

Via this theorem, similar bounds in extremal graph theory can be proven for any excluded subgraph, depending on the chromatic number of the subgraph.

Several choices of the parameter r in a Turán graph lead to notable graphs that have been independently studied.

If n couples go to a party, and each person shakes hands with every person except his or her partner, then this graph describes the set of handshakes that take place; for this reason, it is also called the cocktail party graph.

When r is a divisor of n, the Turán graph is symmetric and strongly regular, although some authors consider Turán graphs to be a trivial case of strong regularity and therefore exclude them from the definition of a strongly regular graph.

The class of Turán graphs can have exponentially many maximal cliques, meaning this class does not have few cliques.

has 3a2b maximal cliques, where 3a + 2b = n and b ≤ 2; each maximal clique is formed by choosing one vertex from each partition subset.

[3] Every Turán graph is a cograph; that is, it can be formed from individual vertices by a sequence of disjoint union and 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.

Chao & Novacky (1982) show that the Turán graphs are chromatically unique: no other graphs have the same chromatic polynomials.

Nikiforov (2005) uses Turán graphs to supply a lower bound for the sum of the kth eigenvalues of a graph and its complement.

[4] 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.

Pór & Wood (2005) give a lower bound of Ω((rn)3/4) on the volume of any three-dimensional grid embedding of the Turán graph.

[6] Witsenhausen (1974) conjectures that the maximum sum of squared distances, among n points with unit diameter in Rd, is attained for a configuration formed by embedding a Turán graph onto the vertices of a regular simplex.

[7] An n-vertex graph G is a subgraph of a Turán graph T(n,r) if and only if G admits an equitable coloring with r colors.

In particular, the Turán graph is the unique maximal n-vertex graph with an r-color equitable coloring.