Free Year 10 Maths Networks & Network Diagrams Worksheet

This worksheet introduces networks and network diagrams, teaching students to find the degree of a vertex, apply the handshake lemma, and determine whether a network has an Eulerian path or circuit. The worksheet is split into warm-up questions, standard questions and then extension questions to test you.

The concepts it covers:

• A network (graph) consists of vertices (nodes/points) connected by edges (lines/connections) — used to model relationships like roads, friendships, or computer connections.
• The degree of a vertex is the number of edges connected to it. The Handshake Lemma: the sum of all vertex degrees always equals twice the number of edges.
• A graph is connected if there is a path between every pair of vertices.
• An Eulerian circuit (visiting every edge exactly once, returning to the start) exists if and only if the graph is connected and every vertex has even degree. An Eulerian path (not returning to start) exists if and only if exactly 0 or 2 vertices have odd degree.
• A minimum spanning tree connects all vertices of a weighted network using the smallest possible total edge weight, with no cycles.

What your child will practise:

• Counting vertices and edges from a network diagram or description, and find the degree of a given vertex.
• Applying the Handshake Lemma to check or find the sum of vertex degrees.
• Determining whether a network is connected.
• Determining whether a network has an Eulerian circuit, an Eulerian path, or neither, using the odd-degree vertex count.
• Finding the shortest path or minimum spanning tree in a small weighted network by comparing candidate routes/trees.

Every section opens with a worked example, and the download includes a full answer key with step-by-step solutions and teaching notes on the mistakes students most commonly make.