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.

$A = \begin{pmatrix} 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \end{pmatrix}$

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 $v_1v_1$ (loop), $v_1v_2$, and $v_2v_3$.

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

$A = \begin{pmatrix} 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix}.$

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 $1 \to 2$, $2 \to 3$, $3 \to 1$, $4 \to 5$, $5 \to 6$, $6 \to 4$, $2 \to 5$.