Lovász theta-function of graphs - Graph Theory

Skip to content
Back to top
View this page
Edit this page

# Lovász theta-function of graphs

AUTHORS:

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

REFERENCE:

[Lov1979]

## 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

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

  Python

      >>> from sage.all import *
      >>> C = graphs.PetersenGraph()
      >>> C.lovasz_theta()                             # optional - csdp
      4.0
      >>> graphs.CycleGraph(Integer(5)).lovasz_theta()          # optional - csdp
      2.236068