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Year 10AC9M10SP02

Networks and Connectedness

Interpret networks and network diagrams used to represent relationships in practical situations and describe connectedness.

A network is a picture of what is joined to what. Where things sit on the page means nothing — only the connections matter. That is the idea most students take longest to accept.

Method

Vertices and edges
A vertex is a thing. An edge is a connection between two things. That is the whole vocabulary.
Degree is how many edges touch a vertex
deg⁡(A)=number of edges at A\deg(A) = \text{number of edges at } A
Count the line ends at the vertex, not the vertices it reaches.
Position carries no meaning
The same network can be drawn a hundred ways. Two drawings are the same network if the same pairs are joined.
Connected means you can get anywhere
A network is connected if there is a path between every pair of vertices. If not, it falls into separate pieces.
The handshake result
∑deg⁡=2E\sum \deg = 2E
Every edge has two ends, so the degrees always add to twice the number of edges. A quick check that you have counted correctly.

Worked example

e.g. Five towns. Roads join A–B, A–C, B–C, B–D and C–D. Town E has no road.\text{Five towns. Roads join A–B, A–C, B–C, B–D and C–D. Town E has no road.}
  1. Draw a vertex for each thing being connected. Place them anywhere with room.
    Five vertices labelled A to E with no connections drawnABCDEFive vertices, no edges yet
  2. Draw one edge for each stated connection. Do not add any others.AB, AC, BC, BD, CDAB,\ AC,\ BC,\ BD,\ CD
    Network of five towns where E has no roadsABCDEE is isolated, so this network is not connected
  3. Count the degree of each vertex by counting line ends.deg⁡A=2, deg⁡B=3, deg⁡C=3, deg⁡D=2, deg⁡E=0\deg A = 2,\ \deg B = 3,\ \deg C = 3,\ \deg D = 2,\ \deg E = 0
  4. Check with the handshake result before going further.2+3+3+2+0=10=2×5✓2+3+3+2+0 = 10 = 2 \times 5 \quad\checkmark
  5. Decide whether it is connected. Try to reach every vertex from one starting point.From A you reach B, C and D — but never E\text{From A you reach B, C and D — but never E}
  6. State the answer in the language of the situation, not just the diagram.The network is not connected: town E cannot be reached by road.\text{The network is not connected: town E cannot be reached by road.}
  7. If edges carry values, label them. A path's total is the sum of its edges.A→C→D=4+3=7A \to C \to D = 4 + 3 = 7
    Weighted network with the shortest path from A to D highlighted64293ABCDShortest route from A to D is 7, not the direct-looking one

Practice

Count line ends for degree, and remember that layout means nothing.

1
A network has 6 edges. What do the degrees add to?\text{A network has 6 edges. What do the degrees add to?}
Answer∑deg⁡=2×6=12\sum \deg = 2 \times 6 = 12
2
Degrees are 3, 3, 2, 2, 1. Can this be a network?\text{Degrees are } 3,\ 3,\ 2,\ 2,\ 1. \text{ Can this be a network?}
Answer∑deg⁡=11, which is odd. No — the total must be even.\sum \deg = 11 \text{, which is odd. No — the total must be even.}
3
A vertex has degree 0. What does that mean in a road network?\text{A vertex has degree 0. What does that mean in a road network?}
AnswerThat town has no roads, so the network cannot be connected.\text{That town has no roads, so the network cannot be connected.}
4
Two students draw the same five towns in different arrangements. Are the networks different?\text{Two students draw the same five towns in different arrangements. Are the networks different?}
AnswerNo. Only the pairs joined matter, never the positions on the page.\text{No. Only the pairs joined matter, never the positions on the page.}
5
From the weighted diagram, find the cost of A→B→D\text{From the weighted diagram, find the cost of } A \to B \to D
Answer6+9=15, so it is dearer than A→C→D=76 + 9 = 15 \text{, so it is dearer than } A \to C \to D = 7
6
A network of 4 vertices has every vertex joined to every other. How many edges?\text{A network of 4 vertices has every vertex joined to every other. How many edges?}
Answer4×32=6\frac{4 \times 3}{2} = 6
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