AS Further Mathematics₇₃₆₆

Understand and use the language of graphs including: vertex, edge, trail, cycle, connected, degree, subgraph, subdivision, multiple edge and loop.

Identify or prove properties of a graph including that a graph is Eulerian,

semi-Eulerian or Hamiltonian.

Understand and use Euler’s formula for connected planar graphs.

Understand and use complete graphs and bipartite graphs, including adjacency matrices and the complement of a graph.

Understand and use simple graphs, simple-connected graphs and trees.

Understand and use the language of networks including: node, arc and weight.

Solve network optimisation problems using spanning trees.

Solve route inspection problems.

Find and interpret upper bounds and lower bounds for the travelling salesperson problem.

Evaluate, modify and refine models which use networks.

Solve constrained optimisation problems via graphical methods.

Construct, represent and interpret a precedence (activity) network using

activity-on-node.

Determine earliest and latest start and finish times for an activity network.

Identify critical activities, critical paths and the float of non-critical activities.

Refine models and understand the implications of possible changes in the context of critical path analysis.

Understand, interpret and construct pay-off matrices.

Find play-safe strategies and the value of the game.

Prove the existence or non-existence of a stable solution.

Identify and make use of dominated strategies.

Find optimal mixed strategies for a game including use of graphical methods.

Understand and use binary operations including use of modular arithmetic and matrix multiplication.

Understand, use and prove the commutativity of a binary operation.

Understand, use and prove the associativity of a binary operation.

Construct a Cayley table for a given set under a given binary operation.