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.

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.

Definition 
Let  be a graph on  vertices. An ordered set of  unit vectors  is called an orthonormal representation of  in , if  and  are orthogonal whenever vertices  and  are not adjacent in :

Clearly, every graph admits an orthonormal representation with : just represent vertices by distinct vectors from the standard basis of . Depending on the graph it might be possible to take  considerably smaller than the number of vertices .

The Lovász number  of graph  is defined as follows:

where  is a unit vector in  and  is an orthonormal representation of  in . Here minimization implicitly is performed also over the dimension , however without loss of generality it suffices to consider . Intuitively, this corresponds to minimizing the half-angle of a rotational cone containing all representing vectors of an orthonormal representation of . If the optimal angle is , then  and  corresponds to the symmetry axis of the cone.

Equivalent expressions 
Let  be a graph on  vertices. Let  range over all  symmetric matrices such that  whenever  or vertices  and  are not adjacent, and let  denote the largest eigenvalue of . Then an alternative way of computing the Lovász number of  is as follows:

The following method is dual to the previous one. Let  range over all  symmetric positive semidefinite matrices such that  whenever vertices  and  are adjacent, and such that the trace (sum of diagonal entries) of  is . Let  be the  matrix of ones. Then

Here,  is just the sum of all entries of .

The Lovász number can be computed also in terms of the complement graph . Let  be a unit vector and  be an orthonormal representation of . Then

Value for well-known graphs 

The Lovász number has been computed for the following graphs:

Properties 

If  denotes the strong graph product of graphs  and , then

If  is the complement of , then

with equality if  is vertex-transitive.

Lovász "sandwich theorem" 

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

where  is the clique number of  (the size of the largest clique) and  is the chromatic number of  (the smallest number of colors needed to color the vertices of  so that no two adjacent vertices receive the same color).

The value of  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.
For perfect graphs, the chromatic number and clique number are equal, and therefore are both equal to . By computing an approximation of  and then rounding to the nearest integer value, the chromatic number and clique number of these graphs can be computed in polynomial time.

Relation to Shannon capacity 
The Shannon capacity of graph  is defined as follows:

where  is the independence number of graph  (the size of a largest independent set of ) and  is the strong graph product of  with itself  times. Clearly, . However, the Lovász number provides an upper bound on the Shannon capacity of graph, hence

For example, let the confusability graph of the channel be , a pentagon. Since the original paper of  it was an open problem to determine the value of . It was first established by  that .

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

To show that this bound is tight, let  be the following orthonormal representation of the pentagon:

and let . By using this choice in the initial definition of Lovász number, we get . Hence, .

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.

Quantum physics

The Lovász number has been generalized for "non-commutative graphs" in the context of quantum communication. The Lovasz number also arises in quantum contextuality in an attempt to explain the power of quantum computers.

See also
Tardos function, a monotone approximation to the Lovász number of the complement graph that can be computed in polynomial time and has been used to prove lower bounds in monotone circuit complexity.

Notes

References

External links 

Graph invariants
Information theory