To find the adjacency matrix $A$ for the graph associated with the given matrix $M$, we need to understand the structure of $M$. The matrix $M$ appears to represent an incidence matrix of a graph, where each row corresponds to an edge, and each column represents a vertex. The values in the matrix indicate whether the vertex is connected to an edge and the nature of the connection:

• $1$: The vertex is the head of the directed edge.
• $-1$: The vertex is the tail of the directed edge.
• $0$: The vertex is not connected to the edge.

We can interpret the graph from this incidence matrix and then construct the adjacency matrix $A$, which shows the connections between vertices. For an adjacency matrix:

• $A[i][j] = 1$ if there is an edge between vertex $i$ and vertex $j$.
• $A[i][j] = 0$ if there is no edge between vertex $i$ and vertex $j$.

Steps:

1. Identify vertices and edges from the incidence matrix $M$.
2. Create the adjacency matrix by checking which vertices are connected by an edge, using the incidence matrix.

Matrix $M$:

\begin{pmatrix} 1 & -1 & 0 & 0 & 0 & 1 & 0 \\ -1 & 1 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & -1 \\ 0 & 0 & 0 & -1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & -1 & -1 & 0 \end{pmatrix}$$ Here: - There are **7 vertices** (columns of the matrix). - There are **5 edges** (rows of the matrix). #### Construction of Adjacency Matrix $$A$$: - From row 1, vertices 1 and 2 are connected, so $$A[1][2] = 1$$ and $$A[2][1] = 1$$. - From row 2, vertices 2 and 3 are connected, so $$A[2][3] = 1$$ and $$A[3][2] = 1$$. - From row 3, vertices 3 and 4 are connected, so $$A[3][4] = 1$$ and $$A[4][3] = 1$$. - From row 4, vertices 4 and 5 are connected, so $$A[4][5] = 1$$ and $$A[5][4] = 1$$. - From row 5, vertices 5 and 6 are connected, so $$A[5][6] = 1$$ and $$A[6][5] = 1$$. Thus, the adjacency matrix $$A$$ for this graph is: $$A = \begin{pmatrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \end{pmatrix}$$ This matrix $$A$$ represents the adjacency matrix of the graph derived from the incidence matrix $$M$$. Would you like more details on how this adjacency matrix was constructed or how incidence matrices work? ### Related Questions: 1. How can we find the degree of each vertex from the adjacency matrix? 2. What is the difference between an incidence matrix and an adjacency matrix? 3. How can we determine if a graph is directed or undirected from its incidence matrix? 4. How does the adjacency matrix change if we add more edges to the graph? 5. What properties of the graph can we derive from the adjacency matrix (e.g., connectivity)? #### Tip: The adjacency matrix is symmetric for undirected graphs, as $$A[i][j] = A[j][i]$$ indicates that the edges do not have direction.