Lovász theta-function of graphs

AUTHORS:

  • Dima Pasechnik (2015-06-30): Initial version

REFERENCE:

[Lovasz1979]László Lovász, “On the Shannon capacity of a graph”, IEEE Trans. Inf. Th. 25(1979), 1-7.

Functions

sage.graphs.lovasz_theta.lovasz_theta(graph)

Return the value of Lovász theta-function of graph

For a graph \(G\) this function is denoted by \(\theta(G)\), and it can be computed in polynomial time. Mathematically, its most important property is the following:

\[\alpha(G)\leq\theta(G)\leq\chi(\overline{G})\]

with \(\alpha(G)\) and \(\chi(\overline{G})\) being, respectively, the maximum size of an independent set set of \(G\) and the chromatic number of the complement \(\overline{G}\) of \(G\).

For more information, see the Wikipedia article Lovász_number.

Note

  • Implemented for undirected graphs only. Use to_undirected to convert a digraph to an undirected graph.
  • This function requires the optional package csdp, which you can install with sage -i csdp.

EXAMPLES:

sage: C=graphs.PetersenGraph()
sage: C.lovasz_theta()                             # optional csdp
4.0
sage: graphs.CycleGraph(5).lovasz_theta()          # optional csdp
2.236068