Search topics

Search

Identify the network a connection matrix describes

Start a set of 8
Step 1 of 9

Decode a matrix without double-counting

Use order W, X, Y, Z. First draw all four vertices. Each entry above the diagonal gives the number of edges for one distinct pair; its mirrored cell records the same edges. Each diagonal entry gives the loop count. This avoids turning two mirrored 2s into four XY edges

\[N=\begin{bmatrix}1&0&0&0\\0&0&2&0\\0&2&0&0\\0&0&0&0\end{bmatrix}\]
Year 11, VCE Units 1 & 2, Chapter 2, 3 difficulty levels, No calculatorBuild a worksheet

From an undirected matrix to a graph

The full vertex order, each off-diagonal pair and the diagonal loop counts are all required

  1. List vertices before edges Create every vertex in the declared order, including any whose row is all zero
  2. Use one triangle for ordinary edges For a symmetric undirected matrix, read each off-diagonal pair once, for example above the diagonal, to avoid drawing each edge twice
  3. Read diagonal loops separately Draw exactly aᵢᵢ loops at vertex i under the once-per-loop diagonal convention
  4. Check by re-encoding Rebuild the matrix from the drawing in the same order. Every entry must agree, including zeros and multiplicities

Worked example

Decode N in order W, X, Y, Z, using one diagonal entry per loop

\[N=\begin{bmatrix}1&0&0&0\\0&0&2&0\\0&2&0&0\\0&0&0&0\end{bmatrix}\]
  1. \(\displaystyle V=\{W,X,Y,Z\}\)Create all four vertices, including isolated Z
  2. \(\displaystyle N_{11}=1\)Draw one loop at W
  3. \(\displaystyle N_{23}=N_{32}=2\)Draw two XY edges, not four. Both entries describe the same pair
  4. \(\displaystyle \begin{gathered}E:\ WW,XY,XY\\|E|=3\end{gathered}\)Every other pair has zero edges. Re-encode the drawing to check the complete matrix

Common mistakes

Double-counting mirrored entries
✕\(\displaystyle 2+2=4\text{ XY edges}\)✓\(\displaystyle 2\text{ XY edges}\)
The symmetric pair records the same two undirected edges
Dropping isolated vertices
✕\(\displaystyle V=\{W,X,Y\}\)✓\(\displaystyle V=\{W,X,Y,Z\}\)
An all-zero row represents an isolated vertex, not a missing vertex
Confusing recognition and construction
✕\(\displaystyle \text{recognise}\Rightarrow\text{draw independently}\)✓\(\displaystyle \text{construction needs its own attempt}\)
A selected diagram is evidence of recognition, not evidence that an entire graph was independently constructed

Difficulty levels

introRecognise the matching small undirected diagram; not independent drawing
standardRead numeric features of a four-vertex undirected matrix
challengeRead numeric features of a five-vertex undirected matrix

Questions about this topic

How can I avoid drawing the same edge twice?+

Read one off-diagonal triangle, for example entries above the main diagonal, and draw the given number of edges for each pair

Do I ignore the diagonal?+

No. Read it separately to draw each loop once under the stated convention

Can a graph with an isolated vertex still have a matrix?+

Yes. Keep its row and column, both zero

Does a correct graph-choice answer grade my drawing?+

No. It grades recognition of a supplied diagram. Full drawing and explanation remain ungraded paper practice with a model

Why are one-way variants absent?+

This section introduces the study-design demand for undirected graphs. The retained generators have directed variants elsewhere, but these are excluded here

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Specialist Mathematics, chapter 2: Graph theory

In this chapter

Count the vertices, edges and loops of a graphRepresent a network in a matrixIdentify the network a connection matrix describesDegrees, the degree sum, and even and odd vertices2B Graph families and structure: Graph types, subgraphs, connection models and method selectionConnected graphs and bridgesIdentify isomorphic graphsTrees and spanning treesGraph families and tree conditionsPlanar graphs and drawings in planar formFaces and Euler's formula2C Planarity and graph proofs: Graph types, subgraphs, connection models and method selectionClassify a route: walk, trail, path, circuit or cycleEulerian trails and circuits: existence and identificationHamiltonian paths and cycles