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<mdoxml version="1.0"><p>A <em>graph</em> is a set of <em>vertices</em> (or <em>nodes</em>), together with a set of <em>edges</em> (or <em>arcs</em>).</p>
<p>Look at the graphs below. You can print them off as a set of cards <a class="pdflink" href="/content/id/8257/Simply%20Graphs.pdf">here</a>.<br></br>
</p>
<table style="width: 600px;border-spacing:1px;" border="1">
<tbody>
<tr>
<td><mdo:image src="Graph1.png"></mdo:image></td>
<td><mdo:image src="Graph2.png"></mdo:image></td>
<td><mdo:image src="Graph3.png"></mdo:image></td>
</tr>
<tr>
<td><mdo:image src="Graph4.png"></mdo:image></td>
<td><mdo:image src="Graph5.png"></mdo:image></td>
<td><mdo:image src="Graph6.png"></mdo:image></td>
</tr>
<tr>
<td><mdo:image src="Graph7.png"></mdo:image></td>
<td><mdo:image src="Graph8.png"></mdo:image></td>
<td><mdo:image src="Graph9.png"></mdo:image></td>
</tr>
<tr>
<td><mdo:image src="Graph10.png"></mdo:image></td>
<td><mdo:image src="Graph11.png"></mdo:image></td>
<td><mdo:image src="Graph12.png"></mdo:image></td>
</tr>
<tr>
<td><mdo:image src="Graph13.png"></mdo:image></td>
<td><mdo:image src="Graph14.png"></mdo:image></td>
<td><mdo:image src="Graph15.png"></mdo:image></td>
</tr>
</tbody>
</table>
<p>Can you find graphs that you think 'belong' together in some sense?<br></br>
Can you describe the features they have in common?<br></br>
Could you draw more graphs that 'belong' in the same set?</p>
<p> </p>
<p>Click below to see some definitions used to describe graphs.</p>
<div class="toggle">
<ul>
<li>A <em>loop</em> is an edge with the same vertex at both ends.</li>
<li>A simple graph has no loops, and there is no more than one edge connecting any pair of vertices. A multigraph has multiple edges connecting some pairs of vertices.</li>
<li>A walk is a sequence of edges in which the end of one edge (except the last) is the beginning of the next.</li>
<li>A trail is a walk in which no edge is repeated.</li>
<li>A path is a trail in which no vertex is repeated.</li>
<li>A cycle is a closed path (the end of the last edge is the start of the first).</li>
<li>A Hamiltonian cycle visits every vertex.</li>
<li>A connected graph has a path between every pair of vertices.</li>
<li>A tree is a simple connected graph with no cycles.</li>
<li>A complete graph is a simple graph where every pair of vertices is connected by an edge.</li>
</ul>
</div>
<p>Can you find examples on the cards that match each definition? Can you draw some more examples of your own?</p></mdoxml>
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<mdoxml version="1.0"><h3>Why do this problem?</h3>
Introductions to graph theory can often end up being quite dry and dusty, with lots of definitions that need to be memorised. This problem invites students to engage with the different types of graph as a pictorial representation so that they can understand why different categorisations are necessary.<br></br>
<br></br>
<h3>Possible approach</h3>
Hand out the <a class="pdflink" href="/content/id/8257/Simply%20Graphs.pdf">cards</a>.<br></br>
"Here is a set of cards. Each card shows a representation of a <em>graph</em>. A graph is a collection of <em>vertices</em>, also called <em>nodes</em>, joined by <em>edges</em>, also called <em>arcs</em>. Sort the cards into groups that you think belong together in some way, and sketch the graphs together with an explanation of why you have sorted them together. See how many different ways of
categorising the graphs you can come up with."<br></br>
<br></br>
While students are working, circulate and look out for groups who have come up with collections of graphs that fit standard definitions. <a class="pdflink" href="/content/id/8257/Simply%20Graphs2.pdf">Here</a> are some definitions that might be appropriate.<br></br>
<br></br>
Bring the class together and share the categorisations they came up with. When they identify a standard categorisation, share the usual terminology with them. Perhaps hand out the definitions sheet and invite them to find examples on the cards for each definition.<br></br>
<br></br>
<h3>Key questions</h3>
What is it about some graphs that make them similar to others?<br></br>
Can you think of different properties of the graphs?<br></br>
Can you trace routes from one node to another?<br></br>
<br></br>
<h3>Possible extension</h3>
<a href="/4719">This article</a> discusses properties of graphs and leads to a proof of Euler's formula.<br></br>
<br></br>
<h3>Possible support</h3>
All students should be able to make a start on this activity by sorting the cards and explaining why they have chosen to group certain graphs together.<br></br></mdoxml>
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<mdoxml version="1.0">What is it about some graphs that make them similar to others?<br></br>
Can you think of different properties of the graphs?<br></br>
Can you trace routes from one node to another?</mdoxml>
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Simply Graphs
Look for the common features in these graphs. Which graphs belong together?
Admin
Stage 5 Decision mapping
Decision Mathematics and Combinatorics
Networks/Graph Theory