The 
-state Potts model on a finite
graph 
assigns to each graph vertex 
a state 
. For a positive
integer 
and real interaction parameter 
, set 
.
The zero-field partition function can then be written
 |
(1)
|
where 
is the Kronecker
delta. Equal states on adjacent graph vertices
are favored when 
and disfavored when 
.
Potts (1952) introduced the model as a 
-state generalization of the Ising
model, which is recovered at 
up to a constant normalization and a rescaling of 
.
Expanding the product and then summing over the states gives the Fortuin-Kasteleyn or random-cluster model representation
 |
(2)
|
where 
is the number of connected components of the
spanning subgraph 
(Fortuin and Kasteleyn 1972). Indeed, after the graph
edges in 
have imposed equality of incident states, each connected
component may be assigned any one of the 
states. The right side is a polynomial
in 
and 
, so it extends the partition function from positive
integers 
to arbitrary complex numbers 
and 
.
For the homogeneous random-cluster model in which every graph edge has parameter 
, the corresponding normalizing sum is
 |
(3)
|
Thus 
gives independent bond
percolation, while other positive values of 
weight a configuration of graph
edges according to its number of connected
components.
Setting 
and 
, the same expansion identifies the
Potts partition function with an evaluation of the Tutte
polynomial:
 |
(4)
|
At 
, adjacent graph
vertices must have different states, and hence
 |
(5)
|
where 
is the chromatic polynomial. Consequently,
the Potts model simultaneously provides state-sum, random-cluster, and graph-polynomial
interpretations of the same invariant of a finite
graph (Wu 1982).
