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.[1] This measure, first introduced by Körner in the 1970s,[2][3] has since also proven itself useful in other settings, including combinatorics.[4]
Definition
Let
be an undirected graph. The graph entropy of
, denoted
is defined as
![{\displaystyle H(G)=\min _{X,Y}I(X;Y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f795ba580f101ffcda174a0705ee59e35634365)
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
.[5]
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 (i) the marginal distribution of the first term is uniform and (ii) 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
- Monotonicity. If
is a subgraph of
on the same vertex set, then
.
- Subadditivity. Given two graphs
and
on the same set of vertices, the graph union
satisfies
.
- Arithmetic mean of disjoint unions. Let
be a sequence of graphs on disjoint sets of vertices, with
vertices, respectively. Then
.
Additionally, simple formulas exist for certain families classes of graphs.
- Complete balanced k-partite graphs have entropy
. In particular,
- Edge-less graphs have entropy
.
- Complete graphs on
vertices have entropy
.
- Complete balanced bipartite graphs have entropy
.
- Complete bipartite graphs with
vertices in one partition and
in the other have entropy
, where
is the binary entropy function.
Example
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.