DP Further Mathematics HL Questionbank

• Syllabus

6.7

Sub sections and their related questions

Graphs, vertices, edges, faces. Adjacent vertices, adjacent edges.

• 11M.1.hl.TZ0.4b: (i)     A simple graph has e edges and v vertices, where \(v > 2\) . Prove that if all the...
• 11M.2.hl.TZ0.1a: Draw a planar graph to represent this map.
• 11M.2.hl.TZ0.1b: Write down the adjacency matrix of the graph.
• 11M.2.hl.TZ0.1c: List the degrees of each of the vertices.
• 10M.1.hl.TZ0.3a: (i)     Write down the adjacency matrix for \(G\) . (ii)     Find the number of walks of length...
• 10M.1.hl.TZ0.3b: (i)     Draw \(G'\), the complement of \(G\) . (ii)     Write down the degrees of all the...
• 13M.2.hl.TZ0.4a: (i)     Draw the planar graph \(H\) that represents these mutual friendships. (ii)     State how...
• 13M.2.hl.TZ0.4c: Verify Euler’s formula for \(H\) .
• 13M.2.hl.TZ0.4d: State, giving a reason, whether or not \(H\) is bipartite.
• 13M.2.hl.TZ0.4e: Write down the adjacency matrix for \(H\) .
• 13M.2.hl.TZ0.4f: David wishes to send a message to Grace, in a sealed envelope, through mutual friends. In how...
• 08M.1.hl.TZ0.1b: (i)     Write down the adjacency matrix for \(G\). (ii)     Find the number of distinct walks of...
• 08M.2.hl.TZ0.1A.a: The graph \(G\) has the following cost adjacency matrix. Draw \(G\) in planar form.
• 12M.2.hl.TZ0.2A.a: (i)     Show that \(H\) is bipartite. (ii)     Draw \(H\) as a planar graph.
• 12M.2.hl.TZ0.2A.d: Find the maximum number of extra edges that can be added to \(H\) while keeping it simple, planar...
• SPNone.2.hl.TZ0.6c: The graph \(P\) has the following adjacency table, defined for vertices A to H, where each...
• 15M.1.hl.TZ0.4a: Draw the simple graph \(G\).
• 18M.2.hl.TZ0.3a: Draw a graph G to represent this floorplan where the rooms are represented by the vertices and an...

Degree of a vertex, degree sequence.

• 18M.2.hl.TZ0.3a: Draw a graph G to represent this floorplan where the rooms are represented by the vertices and an...

Handshaking lemma.

• 18M.2.hl.TZ0.3a: Draw a graph G to represent this floorplan where the rooms are represented by the vertices and an...

Simple graphs; connected graphs; complete graphs; bipartite graphs; planar graphs; trees; weighted graphs, including tabular representation.

• 08M.1.hl.TZ0.1a: Determine whether or not \(G\) is bipartite.
• 07M.2.hl.TZ0.1a: (i)     Explain briefly why \(G\) has no Eulerian circuit. (ii)     Determine whether or not...
• 15M.1.hl.TZ0.4a: Draw the simple graph \(G\).
• 15M.1.hl.TZ0.4d: State whether or not \(G\) is planar, giving a reason for your answer.
• 15M.1.hl.TZ0.4e: State whether or not the simple graph \(G\) is bipartite, giving a reason for your answer.
• 18M.2.hl.TZ0.3a: Draw a graph G to represent this floorplan where the rooms are represented by the vertices and an...

Subgraphs; complements of graphs.

• 15M.1.hl.TZ0.4f: Draw the complement \(G'\) of \(G\).
• 18M.2.hl.TZ0.3a: Draw a graph G to represent this floorplan where the rooms are represented by the vertices and an...

Euler’s relation: \(v - e + f = 2\) ; theorems for planar graphs including \(e \leqslant 3v - 6\) , \(e \leqslant 2v - 4\) , leading to the results that \({\kappa _5}\) and \({\kappa _{3,3}}\) are not planar.

• SPNone.2.hl.TZ0.6a: A connected planar graph has \(e\) edges, \(f\) faces and \(v\) vertices. Prove Euler’s relation,...
• SPNone.2.hl.TZ0.6b: (i)     A simple connected planar graph with \(v\) vertices, where \(v \ge 3\) , has no circuit...
• 18M.2.hl.TZ0.3a: Draw a graph G to represent this floorplan where the rooms are represented by the vertices and an...