Dichroic Polynomial

A polynomial Z_G(q,v) in two variables for abstract graphs. A graph with one graph vertex has Z=q. Adding a graph vertex not attached by any graph edges multiplies the Z by q. Picking a particular graph edge of a graph G, the polynomial for G is defined by adding the polynomial of the graph with that graph edge deleted to v times the polynomial of the graph with that graph edge collapsed to a point.

Setting v=-1 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


where N is the number of graph vertices in G. Dichroic polynomials for some simple graphs are


See also

Idiosyncratic Polynomial, Rank Polynomial, Tutte Polynomial

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

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Dichroic Polynomial

Cite this as:

Weisstein, Eric W. "Dichroic Polynomial." From MathWorld--A Wolfram Web Resource.

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