🕸️ Graphs, Adjacency Matrices + Minimum Spanning Trees
Unit 4 Topics 3 and 4 (construction half) · ~70% of the internal's marks live in networks · week 2 of the internal run
The concept in one sentence: a network is dots joined by lines, and nearly every exam question is either changing costume between the three representations (picture, table, matrix) or choosing the cheapest set of lines that still connects everything.
The thing that makes it click: "two costumes, same information." Draw a graph, build its table, build its matrix, then point out he never learned anything new after the picture.
1 · Prerequisite check (30 seconds)
Draw any 4 dots, join some, ask the degree of one vertex. Then ask him to read one cell of any two-way table (a footy ladder works).
He HAS degree counting (one of his three correct diagnostic answers). Say so out loud before teaching anything, it reframes the whole topic as "you already half-know this". If table-reading wobbles, spend two minutes on any two-way table before touching matrices.
2 · The teaching path (concrete → picture → notation)
Concrete: a real map with towns (the sample's outback towns, or servo runs he knows). Roads = edges, towns = vertices. Ask degree questions on the map, not on an abstract graph
Picture → table: he builds the distance table FROM his own drawing. Symmetry falls out naturally ("A to C is obviously C to A")
Table → matrix: the adjacency matrix is the same table but counting edges instead of distances. Teach the three checks as a chant, and run them EVERY time, even when obviously fine:
symmetric across the diagonal · row sums = degrees · diagonal = loops
MST last: Kruskal as greedy shopping. Sort the edges cheapest first, take each one unless it closes a loop, stop at n − 1 edges. Do the sample's Ms Nguyen table (MST = 30 m) together, then the fresh four-shed table (MST = 15) hands-off
Planar graphs, small but examinable: a graph is planar if it can be redrawn with no edges crossing; a face is a region of the flat drawing, and THE OUTSIDE ALWAYS COUNTS as one. Check the count with the printed formula v + f − e = 2. The IA3 sample asked "how many faces does the graph have" cold, so give it five minutes, not zero
Two costumes, same information. The question is just asking you to change the costume.
Cheapest edge that doesn't close a loop. Say it every time you take one.
3 · Misconception catalogue (⭐ = seen in his diagnostic)
Looks like
Why brains do it
The fix script
⭐ Reaches for critical path when asked for cheapest connection
All network tools blur into one "networks thing"
The which-tool sort: "connect everything cheap = MST, get from A to B = shortest path, schedule a project = critical path". Drill it as a matching game, not an explanation
"A needs its own line" (adds a direct edge when A is already connected through C)
Connected-through doesn't feel connected
Trace the path with a finger: "Can a message get from A to D right now? Then they're connected." Connected THROUGH counts
MST with n edges for n vertices
No self-check habit
Install the ritual: count edges at the end, must be n − 1, every single time. It catches the loop he didn't see
Asymmetric matrix (fills QLD-NSW but not NSW-QLD)
Fills cell by cell reading the map, forgets the mirror
"Every border you find fills TWO cells, same breath." Then the symmetry check catches leftovers
Forgets the isolated vertex's row (Tasmania)
No edges feels like no row
"Every vertex gets a row even if it's all zeros. The zeros ARE the answer"
Reads the weight as the number of edges in a matrix question
Distance tables and adjacency matrices look identical
"Distance table = how far. Adjacency matrix = how many. Check the question's noun"
Runs the symmetry check on a one-way or who-beat-whom question
The chant is drilled as an every-time habit
"One-way streets or round-robin results = a DIGRAPH. Symmetry off, the asymmetry IS the information." (A footy round-robin is the syllabus's own example, expect it)
4 · Questioning ladder
Rung
Ask
Listen for
Recall
"The three matrix checks?"
The chant, unprompted
Do
The four-shed table (AB9, AC4, AD7, BC5, BD8, CD6): draw it, MST it
AC + BC + CD = 15, skips AD with a reason, counts 3 edges for 4 sheds
Explain back
"Why do we skip an edge that closes a loop?"
"It costs money and connects nothing new"
Transfer
"Eight states, build the border matrix" (Australia, no map hints)
Symmetric as he goes, hunts down ACT, volunteers Tasmania's zero row
5 · How QCAA examines it
SF forms: degree and adjacency from a map (Q5 shape), draw the network from a table then MST it (Q6), adjacency matrix + graph from a situation (Q7). These three alone are ~9 marks
The verbs matter: "use a diagram to represent" = they give a blank box, drawing IS the answer. "Identify the minimum spanning tree AND calculate its total length" = two marks, name the edges then add them
CU form: MST with a twist (the walkways question: cost per metre PLUS cost per connection). The reasoning marks come from stating assumptions and justifying with MST language even if the final quote differs
One formula IS printed for networks: Euler's formula v + f − e = 2, and it is for PLANAR graph questions (counting faces of a flat drawing), NOT for Euler trails. The name collision is a known exam wrong-turn, see the Euler guide's catalogue. Everything else in this topic is method, which is why the chant and the ritual matter more here than anywhere
6 · Stuck scripts
Which is the cheapest edge you haven't used yet? Does it close a loop?
Can a message already get there through other towns? Then it's connected.
Run the three checks on what you've got so far.
How many vertices? So how many edges should your tree have?
7 · ADHD delivery notes
Drawing is the most engaging thing this topic offers, let him hold the pen the entire session. You talk, he draws
Colour as he goes: taken edges in one colour, rejected loop-closers crossed out in another. The MST literally emerges in colour
The which-tool confusion responds to drilling, not explaining. The drills page carries the matching questions, point him there rather than re-teaching
Kruskal is genuinely satisfying (greedy + always works). Frame it as "the algorithm that lets you be lazy AND right"