Graph entropy


In information theory, the graph entropy is a measure of the information rate achievable by communicating symbols over a channel in which certain pairs of values may be confused. This measure, first introduced by Körner in the 1970s, has since also proven itself useful in other settings, including combinatorics.

Definition

Let be an undirected graph. The graph entropy of, denoted is defined as
where is chosen uniformly from, ranges over independent sets of G, the joint distribution of and is such that with probability one, and is the mutual information of and.
That is, if we let denote the independent vertex sets in, we wish to find the joint distribution on with the lowest mutual information such that the marginal distribution of the first term is uniform and in samples from the distribution, the second term contains the first term almost surely. The mutual information of and is then called the entropy of.

Properties

Additionally, simple formulas exist for certain families classes of graphs.
Here, we use properties of graph entropy to provide a simple proof that a complete graph on vertices cannot be expressed as the union of fewer than bipartite graphs.
Proof By monotonicity, no bipartite graph can have graph entropy greater than that of a complete bipartite graph, which is bounded by. Thus, by sub-additivity, the union of bipartite graphs cannot have entropy greater than. Now let be a complete graph on vertices. By the properties listed above,. Therefore, the union of fewer than bipartite graphs cannot have the same entropy as, so cannot be expressed as such a union.