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:

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$$. - The diagonal elements are all 0 because there are no self-loops in the graph. Would you like more details or have any questions? ### Further Questions: 1. How do we construct an adjacency matrix for a general bipartite graph $$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.