Question Type 1: Constructing adjacency matrices for a given simple or directed graph Exercises

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

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

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

Construct the adjacency matrix for the directed graph with vertices $\{1,2,3,4\}$ and arcs $1\to2$, $2\to3$, $3\to4$, $4\to1$, $2\to4$, $4\to2$.

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

For the multigraph with vertices $\{A,B,C\}$ having two parallel edges between $A$ and $B$, one edge between $A$ and $C$, and one between $B$ and $C$, write the adjacency matrix.

Construct the adjacency matrix for the undirected graph with vertices $\{v_1,v_2,v_3\}$ and edges $v_1v_1$ (loop), $v_1v_2$, $v_2v_3$.

Construct the adjacency matrix for the directed graph with vertices $\{1,2,3\}$ and arcs $1\to1$ (loop), $1\to2$, $2\to3$, $3\to2$, $3\to3$ (loop).

Construct the adjacency matrix for the directed graph with vertices $\{1,2,3,4,5,6\}$ and arcs $1\to2$, $2\to3$, $3\to1$, $4\to5$, $5\to6$, $6\to4$, $2\to5$.

Given the adjacency matrix of a directed graph

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

Given the directed graph below, construct its adjacency matrix and then write the adjacency matrix of the converse graph (reverse all arcs).

Vertices: $\{A,B,C,D\}$.
Arcs of $G$: $A\to B$, $B\to C$, $C\to A$, $C\to D$.