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# Dichroic Polynomial

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)

Idiosyncratic Polynomial, Rank Polynomial, Tutte Polynomial

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## References

Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 231-235, 1994.

## Referenced on Wolfram|Alpha

Dichroic Polynomial

## Cite this as:

Weisstein, Eric W. "Dichroic Polynomial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DichroicPolynomial.html