https://file.aiquickdraw.com/mathbot/file/1726466150334m110he22.jpg

For question **5a** in the image, we are asked to find \( P(G, \lambda) \) for the graph shown.

### Solution Outline:
\( P(G, \lambda) \) represents the chromatic polynomial of a graph \( G \), which gives the number of ways to color the vertices of the graph using \( \lambda \) colors such that no two adjacent vertices share the same color.

Let's analyze the graph provided in the image:

1. The graph is a **kite graph** or a **diamond shape** with four vertices and five edges, where there are two adjacent triangles.
   
2. The chromatic polynomial \( P(G, \lambda) \) for this graph can be found using standard methods for small graphs, like **deletion-contraction**, **recursion**, or direct analysis.

For a general **kite graph** with four vertices and five edges, the chromatic polynomial is:

\[
P(G, \lambda) = \lambda (\lambda - 1)(\lambda - 2)(\lambda - 2)
\]

This is because:
- The first vertex can be colored in \( \lambda \) ways.
- The second vertex (adjacent to the first) can be colored in \( \lambda - 1 \) ways.
- The third and fourth vertices have fewer options because of the edges connecting them.

Thus, the chromatic polynomial of the graph is:

\[
P(G, \lambda) = \lambda (\lambda - 1)(\lambda - 2)^2
\]

### Conclusion:
The chromatic polynomial \( P(G, \lambda) \) for the given graph is \( \lambda (\lambda - 1)(\lambda - 2)^2 \).

Let me know if you'd like more details on this solution! Here are some related questions for further exploration:

1. What is the chromatic polynomial for a cycle graph with \( n \) vertices?
2. How is the chromatic number of a graph related to its chromatic polynomial?
3. Can a graph with loops have a chromatic polynomial?
4. What is the chromatic polynomial of a complete graph with \( n \) vertices?
5. How can the chromatic polynomial be used to determine if a graph is bipartite?

**Tip**: Always check if your graph contains any special structures like complete subgraphs or bipartite components, as these can simplify the computation of the chromatic polynomial.

Discover how to find the chromatic polynomial P(G, λ) for a kite-shaped graph with 4 vertices and 5 edges. Learn the step-by-step method of calculating graph colorings using standard methods like deletion-contraction and recursion. Ideal for graph theory students.

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