Read a weighted graph practice questions | General Mathematics (Year 11, VCE Units 1 & 2, Graphs and networks, Weighted graphs and shortest paths)

Reading a weighted graph

A weighted graph shows a number on each edge. To find a route total, add the weights of the edges used by that route and keep the stated unit

\text{route total}=\text{sum of edge weights}

Read the edge label The number on a named edge is its weight \(\displaystyle w(A,B)=18\,\text{km}\)
List the route legs Use each pair of consecutive vertices in the stated route \(\displaystyle A\mathord{-}B\mathord{-}C\)
Add every used edge Add each route edge exactly once \(\displaystyle 7+12+9=28\,\text{minutes}\)
Keep the unit State the total using the same unit as the edge weights \(\displaystyle 28\,\text{minutes}\)

Worked example

A delivery route uses the edges P-Q, Q-R and R-S, with weights 11 km, 8 km and 15 km. A separate direct edge P-S has weight 20 km. Find the total length of the stated route P-Q-R-S

P\mathord{-}Q\mathord{-}R\mathord{-}S:\quad 11+8+15\,\text{km}

\(\displaystyle P\mathord{-}Q,\quad Q\mathord{-}R,\quad R\mathord{-}S\)The stated route uses these three consecutive edges
\(\displaystyle 11+8=19\,\text{km}\)Add the first two edge weights
\(\displaystyle 19+15=34\,\text{km}\)Add the final edge weight to the partial total
\(\displaystyle 34\,\text{km}\)The stated route has a total length of 34 kilometres

Common mistakes

Leaving out a route leg

✕\(\displaystyle 11+8=19\,\text{km}\)✓\(\displaystyle 11+8+15=34\,\text{km}\)

A route total must include every edge used by the stated route, so leaving out a leg makes the total too small

Reading a nearby edge

✕\(\displaystyle w(P,Q)=20\,\text{km}\)✓\(\displaystyle w(P,Q)=11\,\text{km}\)

Use the label on the named edge; the 20 km label belongs to the separate direct edge P-S

Using one edge for a whole route

✕\(\displaystyle w(P,S)=20\,\text{km}\)✓\(\displaystyle 11+8+15=34\,\text{km}\)

A direct edge weight describes one edge, whereas the stated route total uses each of its three legs

Difficulty levels

introRead a direct edge and add the weights of a two-leg route

standardAdd a three-leg route through named intermediate vertices

challengeCompare route totals and distinguish the requested edge in a parallel pair

examSelect a route total from values containing a missing-leg or adjacent-edge error

Questions about this topic

What is a weighted graph?+

It is a graph with a numerical weight attached to each edge. In a network, the weights can represent distance, time or cost

How is the length of a route calculated?+

Identify the edges used by consecutive vertices in the route and add their weights once each. Keep the unit used for the edge weights

Does a route with fewer edges always have the smaller total?+

No. A route with more edges can have a smaller total if its edge weights add to less

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, General Mathematics, chapter 8: Graphs and networks

In this chapter

Count the vertices, edges and loops of a graph Degrees, the degree sum, and even and odd vertices Connected graphs and bridges Identify isomorphic graphs Planar graphs and drawings in planar form Faces and Euler's formula 8C Euler's formula: A full network investigation Classify a route: walk, trail, path, circuit or cycle Eulerian trails and circuits: existence and identification 8E Eulerian trails and circuits: A full network investigation Hamiltonian paths and cycles Read a weighted graph The shortest path by inspection 8G Weighted graphs and shortest paths: A full network investigation Trees and spanning trees Minimum spanning trees by Prim's and Kruskal's algorithms 8H Trees and minimum spanning trees: A full network investigation