Let's solve the problem step by step.

Step 1: Understand the Adjacency Matrix

An adjacency matrix $A$ of an undirected graph is a square matrix where each element $A_{ij}$ represents the number of edges (connections) between vertex $i$ and vertex $j$. For an undirected graph:

• $A_{ij} = 1$ if there is an edge between vertices $i$ and $j$.
• $A_{ij} = 0$ if there is no edge between vertices $i$ and $j$.

Step 2: Matrix Multiplication and the $(i, j)$ Entry

Consider the matrix multiplication of $A$ by itself to obtain $A^2$. The $(i, j)$ entry of $A^2$, denoted as $(A^2)_{ij}$, is given by the sum:

$(A^2)_{ij} = \sum_{k=1}^n A_{ik} \cdot A_{kj}$

where $n$ is the number of vertices in the graph, and $k$ ranges over all possible vertices in the graph.

Step 3: Interpretation of $(A^2)_{ij}$

Each term in the sum $A_{ik} \cdot A_{kj}$ corresponds to checking if there is a two-hop path from vertex $i$ to vertex $j$ via an intermediate vertex $k$:

• $A_{ik}$ indicates whether there is an edge from vertex $i$ to vertex $k$.
• $A_{kj}$ indicates whether there is an edge from vertex $k$ to vertex $j$.

Therefore, $A_{ik} \cdot A_{kj} = 1$ if there is an edge from $i$ to $k$ and another edge from $k$ to $j$, meaning there is a path of length 2 from $i$ to $j$ via $k$.

Summing over all $k$, $(A^2)_{ij}$ counts the total number of such two-hop paths from vertex $i$ to vertex $j$.

Conclusion

The $(i, j)$ entry of $A^2$ indeed represents the number of distinct paths of length 2 from vertex $i$ to vertex $j$ in the graph.

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Would you like further details on this topic or have any additional questions?

Here are 5 related questions to consider:

1. What does the adjacency matrix $A^3$ represent in terms of paths in the graph?
2. How would you interpret $(A^2)_{ii}$ in the context of the graph?
3. What properties does $A$ have if the graph is a complete graph?
4. How does the degree of a vertex relate to the adjacency matrix?
5. Can you find a similar interpretation for higher powers of $A$, such as $A^k$?

Tip: When dealing with matrix operations in graph theory, always consider the physical meaning of the operations—such as counting paths, connections, or other structural properties of the graph.