Steps to Solve the Problem:

1. Understanding the Adjacency Matrix:

   • An adjacency matrix represents a graph where the rows and columns correspond to vertices.
   • A 1 in cell $(i, j)$ indicates an edge between vertex $i$ and vertex $j$, and a 0 means no edge.

2. Drawing the Graph:

   • Each row or column corresponds to one of the 8 vertices (labeled $1, 2, \ldots, 8$).
   • Connect vertices wherever there is a 1 in the adjacency matrix.

3. Counting Edges:

   • Count all 1s in the upper triangle of the adjacency matrix (since the graph is undirected, it is symmetric).

4. Calculating the Degree Sequence:

   • The degree of each vertex is the sum of the entries in its corresponding row or column.

Let us compute and draw the graph step by step.

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Adjacency Matrix:

0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 \end{bmatrix}$$ --- ### **Step 1: Count the Edges** - Each edge is represented twice in the matrix (once for each direction). - Count the total number of `1`s in the matrix and divide by 2: - Sum of all entries = $$16$$ - Number of edges = $$\frac{16}{2} = 8$$ --- ### **Step 2: Degree Sequence** - Sum each row of the adjacency matrix to find the degree of each vertex: - Vertex 1: $$1 + 1 + 0 + 1 + 0 + 0 + 0 = 3$$ - Vertex 2: $$1 + 0 + 1 + 0 + 1 + 0 + 0 = 3$$ - Vertex 3: $$1 + 0 + 1 + 0 + 0 + 1 + 0 = 3$$ - Vertex 4: $$0 + 1 + 1 + 0 + 0 + 0 + 1 = 3$$ - Vertex 5: $$1 + 0 + 0 + 0 + 1 + 1 + 0 = 3$$ - Vertex 6: $$0 + 1 + 0 + 0 + 1 + 0 + 1 = 3$$ - Vertex 7: $$0 + 0 + 1 + 0 + 1 + 0 + 1 = 3$$ - Vertex 8: $$0 + 0 + 0 + 1 + 0 + 1 + 1 = 3$$ **Degree sequence:** $$3, 3, 3, 3, 3, 3, 3, 3$$ --- ### **Step 3: Draw the Graph** We will draw the graph based on the adjacency matrix. networkx as nx import matplotlib.pyplot as plt # Define the adjacency matrix adj_matrix = [ [0, 1, 1, 0, 1, 0, 0, 0], [1, 0, 0, 1, 0, 1, 0, 0], [1, 0, 0, 1, 0, 0, 1, 0], [0, 1, 1, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 1, 1, 0], [0, 1, 0, 0, 1, 0, 0, 1], [0, 0, 1, 0, 1, 0, 0, 1], [0, 0, 0, 1, 0, 1, 1, 0], ] # Create the graph from the adjacency matrix G = nx.Graph() for i in range(len(adj_matrix)): for j in range(i, len(adj_matrix)): if adj_matrix[i][j] == 1: G.add_edge(i + 1, j + 1) # Draw the graph plt.figure(figsize=(8, 8)) pos = nx.spring_layout(G) nx.draw(G, pos, with_labels=True, node_color="lightblue", node_size=700, font_size=12, edge_color="gray") plt.title("Graph Representation", fontsize=14) plt.show()### Results: 1. **Number of edges:** $$8$$ 2. **Degree sequence:** $$3, 3, 3, 3, 3, 3, 3, 3$$ 3. **Graph:** Shown above. --- #### Would you like further explanation or assistance with any of the computations? ### Related Questions: 1. What is the importance of a graph's degree sequence? 2. How does the adjacency matrix represent the connections in a graph? 3. Can you identify if a graph is connected using its adjacency matrix? 4. How do different layouts affect the visualization of the graph? 5. What are some real-world applications of graphs represented by adjacency matrices? ### Tip: When analyzing graphs, the **degree sequence** can help you quickly identify properties like regularity, potential symmetry, and whether the graph is Eulerian.