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The chromatic polynomial is a graph polynomial studied in algebraic graph theory, a branch of mathematics. It counts the number of graph colorings as a function of the number of colors and was originally defined by George David Birkhoff to study the four color problem .
Closed formulas for chromatic polynomials are known for many classes of graphs, such as forests, chordal graphs, cycles, wheels, and ladders, so these can be evaluated in polynomial time. If the graph is planar and has low branch-width (or is nonplanar but with a known branch decomposition), then it can be solved in polynomial time using ...
The chromatic symmetric function is a symmetric function invariant of graphs studied in algebraic graph theory, a branch of mathematics. It is the weight generating function for proper graph colorings , and was originally introduced by Richard Stanley as a generalization of the chromatic polynomial of a graph.
Important graph polynomials include: The characteristic polynomial, based on the graph's adjacency matrix. The chromatic polynomial, a polynomial whose values at integer arguments give the number of colorings of the graph with that many colors. The dichromatic polynomial, a 2-variable generalization of the chromatic polynomial
Finally, the third branch of algebraic graph theory concerns algebraic properties of invariants of graphs, and especially the chromatic polynomial, the Tutte polynomial and knot invariants. The chromatic polynomial of a graph, for example, counts the number of its proper vertex colorings.
R. M. Foster had already observed that the chromatic polynomial is one such function, and Tutte began to discover more, including a function f = t(G) counting the number of spanning trees of a graph (also see Kirchhoff's theorem).
The polynomial + + is the Tutte polynomial of the bull graph. The red line shows the intersection with the plane =, which is essentially equivalent to the chromatic polynomial. The Tutte polynomial, also called the dichromate or the Tutte–Whitney polynomial, is a graph polynomial.
Ronald Cedric Read (19 December 1924 – 7 January 2019) was a British mathematician, latterly a professor emeritus of mathematics at the University of Waterloo, Canada.He published many books [1] and papers, primarily on enumeration of graphs, graph isomorphism, chromatic polynomials, and particularly, the use of computers in graph-theoretical research.