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Isomorphic graphs have the same chromatic polynomial, but non-isomorphic graphs can be chromatically equivalent. For example, all trees on n vertices have the same chromatic polynomial. In particular, ( x − 1 ) 3 x {\displaystyle (x-1)^{3}x} is the chromatic polynomial of both the claw graph and the path graph on 4 vertices.
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. [1]
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
The 7 cycles of the wheel graph W 4. For odd values of n, W n is a perfect graph with chromatic number 3: the vertices of the cycle can be given two colors, and the center vertex given a third color. For even n, W n has chromatic number 4, and (when n ≥ 6) is not perfect. W 7 is the only wheel graph that is a unit distance graph in the ...
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 windmill graph has chromatic number k and chromatic index n(k – 1). Its chromatic polynomial can be deduced from the chromatic polynomial of the complete graph and is equal to = (). The windmill graph Wd(k,n) is proved not graceful if k > 5. [3]
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.
The inclusion exclusion principle forms the basis of algorithms for a number of NP-hard graph partitioning problems, such as graph coloring. [11] A well known application of the principle is the construction of the chromatic polynomial of a graph. [12]