Lovász number

In graph theory, the Lovász number of a graph is a real number that is an upper bound on the Shannon capacity of the graph.

It is also known as Lovász theta function and is commonly denoted by

, using a script form of the Greek letter theta to contrast with the upright theta used for Shannon capacity.

This quantity was first introduced by László Lovász in his 1979 paper On the Shannon Capacity of a Graph.

[1] Accurate numerical approximations to this number can be computed in polynomial time by semidefinite programming and the ellipsoid method.

The Lovász number of the complement of any graph is sandwiched between the chromatic number and clique number of the graph, and can be used to compute these numbers on graphs for which they are equal, including perfect graphs.

is called an orthonormal representation of

Clearly, every graph admits an orthonormal representation with

: just represent vertices by distinct vectors from the standard basis of

considerably smaller than the number of vertices

Here minimization implicitly is performed also over the dimension

[3] Intuitively, this corresponds to minimizing the half-angle of a rotational cone containing all representing vectors of an orthonormal representation of

corresponds to the symmetry axis of the cone.

Then an alternative way of computing the Lovász number of

symmetric positive semidefinite matrices such that

are adjacent, and such that the trace (sum of diagonal entries) of

The Lovász number can be computed also in terms of the complement graph

The Lovász number has been computed for the following graphs:[8] If

The Lovász "sandwich theorem" states that the Lovász number always lies between two other numbers that are NP-complete to compute.

so that no two adjacent vertices receive the same color).

can be formulated as a semidefinite program and numerically approximated by the ellipsoid method in time bounded by a polynomial in the number of vertices of G.[12] For perfect graphs, the chromatic number and clique number are equal, and therefore are both equal to

and then rounding to the nearest integer value, the chromatic number and clique number of these graphs can be computed in polynomial time.

(the size of a largest independent set of

However, the Lovász number provides an upper bound on the Shannon capacity of graph,[13] hence

Since the original paper of Shannon (1956) it was an open problem to determine the value of

, since "11", "23", "35", "54", "42" are five mutually non-confusable messages (forming a five-vertex independent set in the strong square of

By using this choice in the initial definition of Lovász number, we get

However, there exist graphs for which the Lovász number and Shannon capacity differ, so the Lovász number cannot in general be used to compute exact Shannon capacities.

[14] The Lovász number has been generalized for "non-commutative graphs" in the context of quantum communication.

[15] The Lovasz number also arises in quantum contextuality[16] in an attempt to explain the power of quantum computers.