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 ϑ. 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.
It is sandwiched between the chromatic number and clique number of any graph, and can be used to compute these numbers on graphs for which they are equal, including perfect graphs.
Definition
Let G = be a graph on n vertices. An ordered set of n unit vectors U = ⊂ RN is called an orthonormal representation of G in RN, if ui and uj are orthogonal whenever vertices i and j are not adjacent in G:Clearly, every graph admits an orthonormal representation with N = n, but in general it might be possible to take N considerably smaller than the number of vertices n.
The Lovász number ϑ of graph G is defined as follows:
where c is a unit vector in RN and U is an orthonormal representation of G in RN. Here minimization implicitly is performed also over the dimension N, however without loss of generality it suffices to consider N = n. Intuitively, this corresponds to minimizing the half-angle of a rotational cone containing all representing vectors of an orthonormal representation of G. If the optimal angle is ϕ, then ϑ = 1/cos2 and c corresponds to the symmetry axis of the cone.
Equivalent expressions
Let G = be a graph on n vertices. Let A range over all n × n symmetric matrices such that aij = 1 whenever i = j or ij ∉ E, and let λmax denote the largest eigenvalue of A. Then an alternative way of computing the Lovász number of G is as follows:The following method is dual to the previous one. Let B range over all n × n symmetric positive semidefinite matrices such that bij = 0 for every ij ∈ E and Tr = 1. Here Tr denotes trace and J is the n × n matrix of ones. Then
Tr is just the sum of all entries of B.
The Lovász number can be computed also in terms of the complement graph. Let d be a unit vector and V = be an orthonormal representation of. Then
Value of ϑ for some well-known graphs
| Graph | Value of ϑ |
| Complete graph | |
| Empty graph | |
| Pentagon graph | |
| Cycle graphs | |
| Petersen graph | |
| Kneser graphs | |
| Complete multipartite graphs |
Properties
If G ⊠ H denotes the strong graph product of graphs G and H, thenIf is the complement of G, then
with equality if G 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 ω is the clique number of G and χ is the chromatic number of G.
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 G is defined as follows:where α is the independence number of graph G and Gk is the strong graph product of G with itself k times. Clearly, Θ ≥ α. 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 C5, a pentagon. Since the original paper of it was an open problem to determine the value of Θ. It was first established by that Θ = .
Clearly, Θ ≥ α = 2. However, α ≥ 5, since "11", "23", "35", "54", "42" are five mutually non-confusable messages, thus Θ ≥ .
To show that this bound is tight, let U = be the following orthonormal representation of the pentagon:
and let c = . By using this choice in the initial definition of Lovász number, we get ϑ ≤ . Hence, Θ = .
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.