In graph theory, a graph is said to be a pseudorandom graph if it obeys certain properties that random graphs obey with high probability. There is no concrete definition of graph pseudorandomness, but there are many reasonable characterizations of pseudorandomness one can consider.

Pseudorandom properties were first formally considered by Andrew Thomason in 1987.-jumbled if

(equivalently, the number of edges in the subgraph induced). It can be shown that the Erdős–Rényi is almost surely-jumbled.-vertex complete graph, making jumbledness a reasonable quantifier for "random-like" properties of a graph's edge distribution.

Connection to local conditions

-jumbled. This result shows how to check the jumbledness condition algorithmically in polynomial time in the number of vertices, and can be used to show pseudorandomness of specific graphs.

Chung–Graham–Wilson theorem

In the spirit of the conditions considered by Thomason and their alternately global and local nature, several weaker conditions were considered by Chung, Graham, and Wilson in 1989: can satisfy each of these conditions if

• Discrepancy.

• Discrepancy on individual sets.

• Subgraph counting.

• 4-cycle counting.

• Codegree,

• Eigenvalue bounding are the eigenvalues of the adjacency matrix.

, using little-o notation.

. A sequence of graphs which satisfies those conditions is called quasi-random. It is considered particularly surprising that the weak condition of having the "correct" 4-cycle density implies the other seemingly much stronger pseudorandomness conditions. Graphs such as the 4-cycle, the density of which in a sequence of graphs is sufficient to test the quasi-randomness of the sequence, are known as forcing graphs.

, and 4-cycle counting is a special case of subgraph counting. In addition, the graph counting lemma, a straightforward generalization of the triangle counting lemma, implies that the discrepancy condition implies subgraph counting.

The fact that 4-cycle counting implies the codegree condition can be proven by a technique similar to the second-moment method. Firstly, the sum of codegrees can be upper-bounded:

Given 4-cycles, the sum of squares of codegrees is bounded:

Therefore, the Cauchy–Schwarz inequality gives

to give the desired bound. A proof that the codegree condition implies the discrepancy condition can be done by a similar, albeit trickier, computation involving the Cauchy–Schwarz inequality.

. The two conditions can then be shown to be equivalent by invocation of the Courant–Fischer theorem.

Connections to graph regularity

The concept of graphs that act like random graphs connects strongly to the concept of graph regularity used in the Szemerédi regularity lemma is called -regular, it holds that

denotes the edge density behave in a "random-like" fashion. In addition, it was shown by Miklós Simonovits and Vera T. Sós in 1991 that a graph satisfies the above weak pseudorandomness conditions used in the Chung–Graham–Wilson theorem if and only if it possesses a Szemerédi partition where nearly all densities are close to the edge density of the whole graph.

Sparse pseudorandomness

Chung–Graham–Wilson theorem analogues

-regular. The following sparse analogues of the discrepancy and eigenvalue bounding conditions are commonly considered:

• Sparse discrepancy.

• Sparse eigenvalue bounding are the eigenvalues of the adjacency matrix.

but is likely to satisfy the discrepancy property. However, David Conlon and Yufei Zhao proved in 2017 that, for vertex-transitive graphs. One direction of this follows from the expander mixing lemma, while the other requires the assumption that the graph is vertex-transitive and uses the Grothendieck inequality.

Consequences of eigenvalue bounding

vertices is called an -graph. The Alon-Boppana bound., which are termed Ramanujan graphs. They have been studied extensively and there are a number of open problems relating to their existence and commonness.

graph:

• , the vertex-connectivity

• edge-connected contains a perfect matching.

• The maximum cut.

• The largest independent subset

• The chromatic number

Connections to the Green–Tao theorem

Pseudorandom graphs factor prominently in the proof of the Green–Tao theorem. The theorem is proven by transferring Szemerédi's theorem, the statement that a set of positive integers with positive natural density in the integers). The transference to sparse sets requires that the sets behave pseudorandomly, in the sense that corresponding graphs and hypergraphs have the correct subgraph densities for some fixed set of small (hyper)subgraphs. It is then shown that a suitable superset of the prime numbers, called pseudoprimes, in which the primes are dense obeys these pseudorandomness conditions, completing the proof.

References