AHL 3.15—Adjacency matrices and tables

Vertex 2 is connected to vertices $1$, $3$, and $4$ according to the edge list.

The sum of row $i$ in an adjacency matrix equals the degree of vertex $i$, i.e. the number of edges incident to it.

Vertex 2 has edges to $1$, $3$, and $4$, giving degree $3$.

The row sum for vertex 2 is $3$.

This undercounts by missing one of the three edges connected to vertex 2.

This overcounts, perhaps by double-counting an edge or including a non-existent self-loop.

This only counts one edge, ignoring that vertex 2 has three connections.

Only $3$ correctly reflects the three edges $1$–$2$, $2$–$3$, $2$–$4$.

The graph has edges $A$–$B$ and $B$–$C$ only, with no edge between $A$ and $C$.

Since the graph is undirected, place $1$ in both $(A,B)$/$(B,A)$ and $(B,C)$/$(C,B)$.

The $(A,C)$ and $(C,A)$ entries are $0$ since there's no edge; diagonal is $0$ for a simple graph.

This gives $\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right]$.

This incorrectly includes an edge between $A$ and $C$, which does not exist.

This has non-zero diagonal entries, implying self-loops which aren't present in a simple graph.

This is not symmetric, treating the graph as directed when it is undirected.

Option A correctly represents only the two existing edges symmetrically.