The adjacency matrix for the complete bipartite graph \( K_{2,3} \) represents the connections between two sets of vertices. In this case, \( K_{2,3} \) consists of two subsets of vertices: one with 2 vertices (say \( A_1 \) and \( A_2 \)) and another with 3 vertices (say \( B_1 \), \( B_2 \), and \( B_3 \)). Each vertex in the first set is connected to every vertex in the second set, and there are no edges within each set.

The adjacency matrix for \( K_{2,3} \) will be a \( 5 \times 5 \) matrix because the graph has 5 vertices in total. Here's the matrix:

\[
\begin{bmatrix}
0 & 0 & 1 & 1 & 1 \\
0 & 0 & 1 & 1 & 1 \\
1 & 1 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0
\end{bmatrix}
\]

### Explanation:
- The first two rows represent the vertices \( A_1 \) and \( A_2 \). They have 1s corresponding to the connections to \( B_1, B_2, \) and \( B_3 \).
- The last three rows represent the vertices \( B_1, B_2, \) and \( B_3 \). They have 1s corresponding to the connections to \( A_1 \) and \( A_2 \).
- The diagonal elements are all 0 because there are no self-loops in the graph.

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### Further Questions:
1. How do we construct an adjacency matrix for a general bipartite graph \( K_{m,n} \)?
2. What are the properties of the graph \( K_{2,3} \), such as chromatic number and diameter?
3. Can the adjacency matrix of \( K_{2,3} \) be used to find the degree of each vertex?
4. How does the adjacency matrix change if we add or remove an edge?
5. What is the incidence matrix for \( K_{2,3} \) and how does it differ from the adjacency matrix?

### Tip:
In a bipartite graph like \( K_{m,n} \), no two vertices within the same subset are connected, which is why the corresponding submatrices in the adjacency matrix are all zeros.

Explore the adjacency matrix for the complete bipartite graph K2,3. This solution covers key graph theory concepts, including how to construct the adjacency matrix for bipartite graphs and the properties of K2,3. Suitable for students in Grades 11-12 and undergraduate levels.

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