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 has three consecutive legs and a separate direct edge. Find the total length of the three-leg route
\[P\mathord{-}Q:11\,\text{km},\quad Q\mathord{-}R:8\,\text{km},\quad R\mathord{-}S:15\,\text{km},\quad P\mathord{-}S:20\,\text{km};\quad P\mathord{-}Q\mathord{-}R\mathord{-}S\]
- \(\displaystyle P\mathord{-}Q,\quad Q\mathord{-}R,\quad R\mathord{-}S\)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 smallReading 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 legsDifficulty 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 12, VCE Units 3 & 4, General Mathematics, chapter 13: Graphs and networks
In this chapter
Count the vertices, edges and loops of a graphDegrees, the degree sum, and even and odd verticesGraph types, subgraphs, connection models and method selectionConnected graphs and bridgesIdentify isomorphic graphsPlanar graphs and drawings in planar formFaces and Euler's formulaRepresent a network in a matrixIdentify the network a connection matrix describesClassify a route: walk, trail, path, circuit or cycleEulerian trails and circuits: existence and identificationHamiltonian paths and cyclesRead a weighted graphThe shortest path by inspectionDijkstra's algorithm for shortest distances and routesTrees and spanning treesMinimum spanning trees by Prim's and Kruskal's algorithmsA full network investigation