The **chromatic polynomial** is a polynomial that encodes the number of ways to color the vertices of a graph using a given number of colors such that no two adjacent vertices share the same color. Specifically, for a graph \( G \), the chromatic polynomial \( P(G, k) \) is defined as the number of ways to color the graph with \( k \) colors. ### Key Properties 1.
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For each positive integer , the chromatic polynomial counts the proper vertex colourings of a finite graph using a fixed palette of colours.