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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.
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).
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 Euclidean plane. [4]
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.
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 .
As the name indicates, for a given G the function is indeed a polynomial in t. For the example graph, P(G, t) = t(t − 1) 2 (t − 2), and indeed P(G, 4) = 72. The chromatic polynomial includes more information about the colorability of G than does the chromatic number.
If such a k-coloring exists, then we refer to the smallest k needed in order to properly color our graph as the chromatic number, denoted by χ(G). [2] The number of proper k -colorings is a polynomial function of k called the chromatic polynomial of our graph G (by analogy with the chromatic polynomial of undirected graphs) and can be denoted ...
For example, if the events are independent and ... A well known application of the principle is the construction of the chromatic polynomial of a graph. [12]