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			Home &gt; Graph Theory: Networking
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					&gt; Directed Graphs</div>
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        <a class="toc_link" href="">Routing</a>
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        <a class="toc_link" href="page21.xml">Designing A Routing Protocol</a>
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    <div id="content"><h1>Directed Graphs</h1>
				<p>
				Many applications of graph theory require a definition of a graph for
				which edges are <i>directed</i>: allowing there to be an edge
				from <m:math><m:mi>u</m:mi></m:math> to <m:math><m:mi>v</m:mi></m:math> but
				<b>not necessarily</b> an edge from
				<m:math><m:mi>v</m:mi></m:math> to <m:math><m:mi>u</m:mi></m:math>.
				</p>
				
				<p>
				Here is a formal definition:
				</p>
				<p>
				A <i>directed graph</i> (often called a
				<i>digraph</i> for brevity's sake)
				<m:math><m:mi>G</m:mi></m:math> is a pair
				<m:math><m:mfenced><m:mi>V</m:mi><m:mi>E</m:mi></m:mfenced></m:math> where
				<m:math><m:mi>V</m:mi></m:math> is a set of elements called vertices and
				<m:math><m:mi>E</m:mi></m:math> is a subset of the set of all ordered pairs
				<m:math><m:mfenced><m:mi>u</m:mi><m:mi>v</m:mi></m:mfenced></m:math>, where
				<m:math><m:mi>u</m:mi></m:math> and <m:math><m:mi>v</m:mi></m:math> are vertices. (An
				element of <m:math><m:mi>E</m:mi></m:math> is called an edge of
				<m:math><m:mi>G</m:mi></m:math>). For more information on edges and vertices
				please click <a href="../../graph_theory/html/ev_theory.html">here</a>
				</p>
				
				<div class="box">
					<b><i>Example</i></b>
					
						<div class="figure" align="center"><a name=""/><img src="../graphics/simpledigraph1.gif"/><br/><b><i>Figure 2</i></b><br/><br/></div>
						<p>
						A directed graph with
						<m:math>
							<m:mi>V</m:mi>
							<m:mo>=</m:mo>
							<m:mfenced close="}" open="{">
								<m:mi>a</m:mi>
								<m:mi>b</m:mi>
								<m:mi>c</m:mi>
								<m:mi>d</m:mi>
							</m:mfenced>
						</m:math> and 
						<m:math>
							<m:mi>E</m:mi>
							<m:mo>=</m:mo>
							<m:mfenced close="}" open="{">
								<m:mfenced>
									<m:mi>a</m:mi>
									<m:mi>b</m:mi>
								</m:mfenced>
								<m:mfenced>
									<m:mi>a</m:mi>
									<m:mi>c</m:mi>
								</m:mfenced>
								<m:mfenced>
									<m:mi>b</m:mi>
									<m:mi>c</m:mi>
								</m:mfenced>
								<m:mfenced>
									<m:mi>d</m:mi>
									<m:mi>c</m:mi>
								</m:mfenced>
							</m:mfenced>
						</m:math> can be drawn as shown here.
						</p>
					
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