A dichroic polynomial is a polynomial 
in two variables for abstract graphs.
A graph with one graph vertex
has 
. Adding a graph vertex
not attached by any graph edges multiplies the 
by 
. Picking a particular graph
edge of a graph 
, the polynomial
for 
is defined by adding the polynomial of the graph
with that graph edge deleted to 
times the polynomial of the
graph with that graph edge collapsed to a point.
Setting 
gives the chromatic number of the graph.
The dichroic polynomial of a planar graph can be
expressed as the square bracket polynomial
of the corresponding alternating link by
 |
(1)
|
where 
is the number of graph vertices in 
. Dichroic polynomials for some simple graphs
are
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
