Question 23
A network graph is shown.
State the degree of vertex E.
[1 mark]State the number of edges joining D and E.
[1 mark]Construct an adjacency matrix from the graph with the vertices in alphabetical order.
[3 marks]Count edge ends for the degree, edges for the matrix
Watch the edges at E get counted one by one, then watch the matrix fill in a row at a time as each vertex’s edges light up.
Degree counts the edges that meet a vertex, so a loop adds 2. The matrix counts the edges joining each pair, so a loop is 1 and an isolated vertex still gets its row.
Count what meets E, then fill the matrix while the graph lights up
Answer parts a) and b) first. Then build the matrix for part c). Each click on a cell steps it through 0, 1 and 2, and lights up the edges that cell counts. Press Check when every cell has an entry.
Count every edge that meets E, including both arcs to D.
You counted D once. Two separate edges join D and E, and each one adds 1 to the degree of E, so the degree is .
That’s it. One edge to A, one to B and two to D give a degree of .
is the number of edges in the whole graph. The degree of E counts only the edges that meet E: one to A, one to B and two to D.
Look closely at the lens shape between D and E.
Look again at the lens. One arc curves above and one below, and each arc is its own edge. Edges like these, joining the same pair of vertices, are multiple edges, so there are .
That’s it. One arc above, one below, so edges join D and E. Carry that number into part c).
Two quick marks, then three for the matrix
Try each step yourself before you reveal it. Parts a) and b) are one mark each. Part c) gives a mark for one correct row or column, a mark for the loop, and a mark for the whole matrix.
The degree of a vertex is the number of edges that meet it. At E there is one edge to A, one to B and two separate edges to D.
$$\deg(E)=1+1+2=4$$
Writing “Degree = 4”, as the sample response does, is all this mark needs.
Marker: correctly states the degree
The lens between D and E is two arcs, one above and one below. Each arc is a separate edge.
2 edges join D and E.
This is the same 2 that went into the degree of E in part a), and it goes into the matrix in part c).
Marker: correctly states the number of edges
Set up a 5 × 5 matrix with rows and columns in the order A, B, C, D, E. C has no edges, but it is still a vertex, so it keeps its row and column. Then fill a row you can read straight off the graph. Row E counts the edges from E to each vertex in turn.
$$\text{Row E:}\quad\begin{bmatrix}1&1&0&2&0\end{bmatrix}$$
E joins A once, B once, C never, D twice and itself never. The guide notes this mark may be implied by subsequent working, so a fully correct matrix earns it without a separate row.
Marker: correctly completes entries for one row or one column in a 5 × 5 matrix
The loop at A is one edge that starts and finishes at A. The matrix counts edges, so the entry in row A, column A is 1.
$$a_{AA}=1$$
The temptation is to write 2, because a loop adds 2 to the degree of A. Both of its ends meet A, so $\deg(A)=2+1+1=4$. The degree counts edge ends and the matrix counts edges, which is why the guide gives this entry its own mark.
Marker: correctly enters 1 for number of edges joining A to A
Fill the remaining rows the same way, one vertex at a time.
Two checks before you move on. The matrix is symmetric about the leading diagonal, because an edge joining X and Y is counted in row X and in row Y. And row C and column C are all zeros. The guide accepts equivalent matrices, and follow-through applies, so a slip in an earlier entry doesn’t stop you earning this mark if the rest is consistent with the graph.
Marker: completes adjacency matrix
Each phrase of the question gave you something
Reading the degree of a vertex and constructing an adjacency matrix are standard Unit 4 network skills, and the 2023 subject report listed both among the things students did well. The graph is small and every part reads straight off the drawing, with no calculation beyond counting.
If I had to guess where you might lose marks, it would be the three unusual features in the drawing. The loop at A goes into the matrix as 1, and the guide gives that entry a mark of its own. The isolated vertex C still needs a row and a column of zeros, or your matrix is no longer 5 × 5. And the lens between D and E is two edges, which feeds part a), part b) and two cells of the matrix. Count them once carefully and the whole question follows.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording and marking-guide steps are from the 2023 QCAA General Mathematics external assessment, © State of Queensland (QCAA) 2023, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.
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