Use vertex, edge, degree, loop and multiple-edge terminology, apply the handshake rule, and represent a graph with an adjacency matrix.

What this dot point is asking

You must use the vocabulary precisely, apply the handshake rule to check or complete a graph, and convert between a drawn graph and its adjacency matrix.

Core terminology

A graph is a set of vertices (nodes) joined by edges (lines). Key terms:

• Degree of a vertex: the number of edge ends meeting at it.
• Loop: an edge joining a vertex to itself; it adds $2$ to that vertex's degree.
• Multiple edges: two or more edges joining the same pair of vertices.
• Simple graph: a graph with no loops and no multiple edges.
• Connected graph: one in which you can travel between any two vertices.

The adjacency matrix

An adjacency matrix is a square table with one row and one column per vertex. Each entry records the number of edges directly joining that pair.

• For an undirected graph the matrix is symmetric (entry from A to B equals entry from B to A).
• A loop is recorded as a $2$ on the diagonal, matching its contribution of $2$ to the degree.
• The degree of a vertex is the sum of its row (counting a diagonal loop entry once as written).

Why the matrix matters

The adjacency matrix lets you store a network for computation, check the degree sequence quickly, and (by matrix multiplication, met later) count walks of a given length. For directed graphs the matrix is generally not symmetric, which is the first sign that direction matters.