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