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    <div id="titlebar">Graph Theory: Industrial Drilling</div>
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			Home &gt; Graph Theory: Industrial Drilling
					&gt; Foundations
					&gt; A First Theorem</div>
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        <a class="toc_link" href="page1.xml">The Drilling Problem</a>
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        <a class="toc_link" href="page10.xml">A First Theorem</a>
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    <div id="content"><h1>A First Theorem</h1>
				<h2>Your First Theorem in Graph Theory</h2>
				<h2>Theorem:</h2>
				<p>
					Let <m:math><m:mi>G</m:mi></m:math> be a <m:math><m:mi>p</m:mi><m:mo>,</m:mo><m:mi>q</m:mi></m:math>-graph with 
					<m:math>
						<m:mi>V</m:mi>
						<m:mfenced>
							<m:mi>G</m:mi>
						</m:mfenced>
						<m:mo>=</m:mo>
						<m:mrow>
							<m:msub>
								<m:mi>v</m:mi>
								<m:mn>1</m:mn>
							</m:msub>
							<m:mo>,</m:mo>
							<m:msub>
								<m:mi>v</m:mi>
								<m:mn>2</m:mn>
							</m:msub>
							<m:mo>,</m:mo>
							<m:mo>…</m:mo>
							<m:mo>,</m:mo>
							<m:msub>
								<m:mi>v</m:mi>
								<m:mi>p</m:mi>
							</m:msub>
						</m:mrow>
					</m:math> and
					<m:math>
						<m:mo>|</m:mo>
						<m:mi>E</m:mi>
						<m:mo>|</m:mo>
						<m:mo>=</m:mo>
						<m:mi>q</m:mi>
					</m:math>. Then
					<m:math>
						<m:munderover>
							<m:mo>∑</m:mo>
							<m:mrow>
								<m:mi>i</m:mi>
								<m:mo>=</m:mo>
								<m:mn>1</m:mn>
							</m:mrow>
							<m:mi>p</m:mi>
						</m:munderover>
						<m:mi>deg</m:mi>
						<m:mfenced>
							<m:msub>
								<m:mi>v</m:mi>
								<m:mi>i</m:mi>
							</m:msub>
						</m:mfenced>
						<m:mo>=</m:mo>
						<m:mn>2</m:mn>
						<m:mi>q</m:mi>
					</m:math>
				</p>
				<h2>Proof:</h2>
				<p>
				Imagine that every vertex of a graph sends a &quot;message&quot; along every edge incident to it. 
				Messages sent from opposite ends of the same edge collide in the middle of all edges.
				</p>
				
					<p><div class="problem"><a name="problem5"/><b>Problem
				5:
			</b><br/>
						
							<p>
								How many messages are sent?
							</p>
						
						<a target="_new" href="answer2.xml">Answer</a>
					</div></p>
				
				
					<p><div class="problem"><a name="problem6"/><b>Problem
				6:
			</b><br/>
						
							<p>
								How many collisions will occur?
							</p>
						
						<a target="_new" href="answer3.xml">Answer</a>
					</div></p>
				
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					<b><i>Conclusion</i></b>
					
					<p>
					Since the number of collisions is half the number of messages, the theorem is proven.
					</p>
					
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