IB Mathematics Applications & Interpretation Question Type 1: Constructing Adjacency Matrices for a Given Simple or Directed Graph Exercises

Construct the adjacency matrix for the directed graph with vertices $\{A,B,C,D,E\}$ and arcs $A\to B$, $A\to C$, $B\to C$, $C\to A$, $D\to E$.

Given the adjacency matrix $A$ for an undirected simple graph with vertices $1,2,3,4$, determine the total number of edges in the graph.

$$A=\left(\begin{array}{cccc}0& 1& 0& 1\\ 1& 0& 1& 1\\ 0& 1& 0& 0\\ 1& 1& 0& 0\end{array}\right)$$

Construct the adjacency matrix for the undirected graph on vertices $\{1,2,3,4,5\}$ with edges $\{1,2\},\{1,3\},\{1,5\},\{2,3\},\{2,4\},\{3,4\},\{4,5\}$.

Construct the adjacency matrix for the directed graph with vertices $\{1,2,3\}$ and arcs $1\to 1$ (loop), $1\to 2$, $2\to 3$, $3\to 2$, and $3\to 3$ (loop).

A multigraph has vertices $\{A,B,C\}$. There are two edges between $A$ and $B$, one edge between $A$ and $C$, and one edge between $B$ and $C$. There are no other edges and no loops.

State the adjacency matrix for this multigraph, using the order of vertices $(A,B,C)$ for the rows and columns.

Construct the adjacency matrix for the undirected graph with vertex set $\{A,B,C,D\}$ and edges $AB$, $BC$, $CD$, $DA$.

Construct the adjacency matrix for the undirected graph with vertices $\{v_1,v_2,v_3\}$ and edges $v_1{v}_1$ (loop), $v_1{v}_2$, and $v_2{v}_3$.

Construct the adjacency matrix $A_G$ for graph $G$ using the vertex ordering $A,B,C,D$. Then, determine the adjacency matrix $A_{G^{\text{rev}}}$ for the converse graph $G^{\text{rev}}$ (where all arcs are reversed).

Construct the adjacency matrix $\mathbf{A}$ for the directed graph with vertices $\{1,2,3,4\}$ and arcs $1\to 2$, $2\to 3$, $3\to 4$, $4\to 1$, $2\to 4$, $4\to 2$.

The adjacency matrix $A$ of a directed graph is given by

$$A=\left(\begin{array}{cccc}0& 1& 0& 1\\ 0& 0& 1& 0\\ 1& 0& 0& 0\\ 0& 0& 1& 0\end{array}\right).$$

Compute the number of walks of length 2 from vertex 1 to vertex 3 by finding the $(1,3)$ entry of $A^2$.

Construct the adjacency matrix for the directed graph with vertices $\{1,2,3,4,5,6\}$ and arcs $1\to 2$, $2\to 3$, $3\to 1$, $4\to 5$, $5\to 6$, $6\to 4$, $2\to 5$.