NSWMaths Standard 2Syllabus dot point

What are the basic terms and components of a network, and how are real-world systems represented as graphs?

Use network terminology including vertex, edge, weight, degree, path, cycle and directed graph to describe and analyse networks

A focused answer to the HSC Maths Standard 2 dot point on network terminology. Vertices, edges, weights, directed and undirected graphs, paths and cycles, connected and disconnected components, and worked Australian examples for transport networks and project schedules.

Generated by Claude OpusReviewed by Better Tuition Academy7 min answerUpdated 2026-05-20

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What this dot point is asking

NESA wants you to use the standard graph-theory vocabulary (vertex, edge, weight, degree, path, cycle, directed, connected), read a network diagram, and answer basic questions about its structure.

The answer

Basic terminology

Vertex (or node). A point in the network. Usually drawn as a labelled circle.
Edge (or arc). A line connecting two vertices. Edges may be undirected (a two-way connection) or directed (a one-way arrow).
Weight. A number on an edge representing distance, time, cost, capacity, etc.
Degree of a vertex. The number of edges connected to it.
In-degree and out-degree (for directed graphs). The number of edges pointing in to or out from the vertex.

Types of graphs

Undirected graph. Edges have no direction. A road that can be travelled both ways.
Directed graph (digraph). Edges have direction (drawn as arrows). A one-way street, or a task that must be completed before another.
Weighted graph. Each edge has a number attached (distance, time, cost).
Simple graph. No multiple edges between the same pair of vertices, and no loops (an edge from a vertex to itself).

Paths and cycles

Walk. A sequence of vertices connected by edges (possibly revisiting).
Path. A walk in which no vertex is repeated.
Cycle. A path that starts and ends at the same vertex (with at least one intermediate vertex).
Length of a path or cycle. Sum of the weights of its edges (for weighted graphs) or the number of edges (for unweighted).

Connectedness

Connected graph. Every pair of vertices has at least one path between them.
Disconnected graph. Some vertices have no path between them.
Component. A maximal connected subgraph. A disconnected graph has multiple components.

Sum of degrees rule

For any graph (directed or undirected):

\sum_{\text{vertices}} \text{degree} = 2 \times \text{number of edges}.

This is because every edge contributes one to each of its two endpoints. A useful sanity check: the sum is always even, and equals exactly twice the edge count.

For directed graphs, separately:

\sum \text{in-degree} = \sum \text{out-degree} = \text{number of directed edges}.

Representing a graph

Three standard ways:

Vertex-edge diagram. Circles with lines between them, with weights on the edges if applicable.
Adjacency table. A grid showing whether each pair of vertices is connected.
Weighted matrix. A grid showing the weight of the edge between each pair (or blank if no edge).

The HSC usually gives you a diagram; you may need to convert to a table or matrix.

Worked example

Sydney rail network

A simplified Sydney Trains network has stations $W$ (Wynyard), $T$ (Town Hall), $C$ (Central), $R$ (Redfern), $G$ (Granville), $P$ (Parramatta) as vertices. Edges are direct train sections with travel times (in minutes) as weights.

Imagine edges $WT (3)$ , $TC (3)$ , $CR (3)$ , $RG (15)$ , $GP (5)$ .

This is a connected undirected weighted graph. Total edges: $5$ . Sum of degrees: $W: 1$ , $T: 2$ , $C: 2$ , $R: 2$ , $G: 2$ , $P: 1$ , total $10 = 2 \times 5$ . Check.

One-way road network

A small Australian town has roads represented as a directed graph with vertices labelled $A$ - $F$ .

In-degree of $A$ = number of edges arriving at $A$ . Out-degree of $A$ = number of edges leaving $A$ .

A vertex with in-degree higher than out-degree is a "sink-like" location; cars tend to accumulate. A vertex with out-degree higher than in-degree is a "source-like" location; traffic disperses.

Adjacency table

For an undirected graph with vertices $A$ , $B$ , $C$ , $D$ and edges $AB$ , $AC$ , $BD$ , $CD$ :

	A	B	C	D
A	IMATH_33	IMATH_34	IMATH_35	IMATH_36
B	IMATH_37	IMATH_38	IMATH_39	IMATH_40
C	IMATH_41	IMATH_42	IMATH_43	IMATH_44
D	IMATH_45	IMATH_46	IMATH_47	IMATH_48

Symmetric because the graph is undirected. Each row sums to the degree of that vertex.

Project schedule (preview of critical path analysis)

A project has tasks $A$ , $B$ , $C$ , $D$ , $E$ . Some tasks must be completed before others can begin. This is a directed graph (precedence diagram). The vertices are tasks; the directed edges show the order. The graph must be acyclic (no cycles), because a cycle would mean tasks circularly depend on each other.

Past exam questions, worked

Real questions from past NESA papers on this dot point, with our answer explainer.

2022 HSC Q92 marksA network has

6

vertices and

9

edges. Find the sum of the degrees of all vertices, and explain why this must be even.

Show worked answer →

Sum of degrees = $2 \times$ number of edges = $2 \times 9 = 18$ .

The sum of the degrees of all vertices in any graph equals twice the number of edges, because every edge contributes $1$ to each of its two endpoints. So the sum is always even.

Markers reward the formula, the calculation, and the explanation that each edge is counted twice.

2023 HSC Q113 marksA directed graph represents a one-way road network in a small Australian town. List the vertices that have an in-degree greater than out-degree, given vertices labelled

A, B, C, D, E, F

and a description of the directed edges.