## What is the chromatic formula?

If G is a simple graph, we write PG(k) as the number of ways we can achieve a proper coloring on the vertices of G given k colors and PG is called the Chromatic Function of G. If k<χ(G), then PG(k) = 0.

**How do you find the chromatic index of a graph?**

The chromatic index χ ΄ (x) is the minimum number of different colors needed to color edges such that any two adjacent edges are colored by different colors (for more details, see [1, 3,4,5, 7–9, 11,12,13,14]). Kӧnig has proved, in 1916, that χ ΄ (x) = ∆(x) for every bipartite graph.

### What is the chromatic polynomial of C5?

Pk(C5) = k(k − 1)4 − k(k − 1)3 + k(k − 1)(k − 2). Finally, the chromatic polynomial of the graph in the question is Pk(C5) − k(k − 1)2(k − 2) = k(k − 1)4 − k(k − 1)3 + k(k − 1)(k − 2) − k(k − 1)2(k − 2). Let us color the vertices along the path, one by one, starting from one end.

**What is the chromatic polynomial of K3?**

Chromatic polynomial for K3, 3 is given by λ(λ – 1)5. Thus chromatic number of this graph is 2. Since λ(λ – 1)5 > 0 first when λ = 2. Here, only two distinct colours are required to colour K3, 3.

## What is chromatic index of a graph?

The minimum required number of colors for the edges of a given graph is called the chromatic index of the graph. For example, the edges of the graph in the illustration can be colored by three colors but cannot be colored by two colors, so the graph shown has chromatic index three.

**What is the chromatic number of a wheel graph?**

A wheel graph with n vertices can also be defined as the 1-skeleton of an (n-1)-gonal pyramid….

Wheel graph | |
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Chromatic number | 4 if n is even 3 if n is odd |

Spectrum | |

Properties | Hamiltonian Self-dual Planar |

Notation | Wn |

### What is the chromatic polynomial of a cycle?

Proposition The chromatic polynomial of the cycle of size n is (k-1) +(-1) (k-1). Proof We proceed by induction. Base case: n=2. This is the complete graph (plus a multiple edge), so the chromatic polynomial is k(k-1).

**What is chromatic polynomial with example?**

The chromatic polynomial is a graph polynomial studied in algebraic graph theory, a branch of mathematics….Algorithms.

Chromatic polynomial | |
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Input | Graph G with n vertices. |

Output | Coefficients of |

Running time | for some constant |

Complexity | #P-hard |

## What is chromatic number of a graph explain with example?

The chromatic number, χ(G), of a graph G is the smallest number of colors for V(G) so that adjacent vertices are colored differently. Def. 8-2. The chromatic number, χ(Sk),of a surface Sk is the largest χ(G) such that G can be imbedded in Sk. We prove that six colors will suffice for every planar graph.

**What is chromatic index and chromatic number of a graph?**

The edge chromatic number, sometimes also called the chromatic index, of a graph is fewest number of colors necessary to color each edge of. such that no two edges incident on the same vertex have the same color. In other words, it is the number of distinct colors in a minimum edge coloring.