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    <div id="content"><h1>Definitions</h1>
				<h2>Definitions Associated With Directed Graphs</h2>
				<p>
				Digraphs are especially important in networking, where the main
				objects are collections of nodes with one-way links between them.
				</p>
				<p>
				Many <a href="../../graph_theory/html/crib.html">definitions from undirected
				graph theory</a> have digraph-analogues:
				</p>
				<ol><li>
					The <i>out-degree</i> of a vertex
					<m:math><m:mi>u</m:mi></m:math> is the number of edges
					<m:math>
						<m:mfenced>
							<m:mi>u</m:mi>
							<m:mi>v</m:mi>
						</m:mfenced>
					</m:math>
					for all
					<m:math>
						<m:mi>v</m:mi>
						<m:mo>∈</m:mo>
						<m:mi>V</m:mi>
						<m:mo>⁡</m:mo>
						<m:mfenced>
							<m:mi>G</m:mi>
						</m:mfenced>
					</m:math>.
					</li><li>
					The <i>in-degree</i> of a vertex
					<m:math><m:mi>u</m:mi></m:math>
					is the number of edges
					<m:math>
						<m:mfenced>
							<m:mi>v</m:mi>
							<m:mi>u</m:mi>
						</m:mfenced>
					</m:math>
					for all
					<m:math>
						<m:mi>v</m:mi>
						<m:mo>∈</m:mo>
						<m:mi>V</m:mi>
						<m:mo>⁡</m:mo>
						<m:mfenced>
							<m:mi>G</m:mi>
						</m:mfenced>
					</m:math>.
					</li><li>
					Recall that in undirected graphs, a <i>walk</i> from vertex
					<m:math><m:mi>v</m:mi></m:math> to a vertex <m:math><m:mi>u</m:mi></m:math> is an
					alternating finite sequence of vertices and edges
						<m:math>
							<m:msub>
								<m:mi>v</m:mi>
								<m:mn>0</m:mn>
							</m:msub>
							<m:msub>
								<m:mi>e</m:mi>
								<m:mn>0</m:mn>
							</m:msub>
							<m:msub>
								<m:mi>v</m:mi>
								<m:mn>1</m:mn>
							</m:msub>
							<m:msub>
								<m:mi>e</m:mi>
								<m:mn>1</m:mn>
							</m:msub>
							<m:mo>…</m:mo>
							<m:msub>
								<m:mi>v</m:mi>
								<m:mrow>
									<m:mi>k</m:mi>
									<m:mo>-</m:mo>
									<m:mn>1</m:mn>
								</m:mrow>
							</m:msub>
							<m:msub>
								<m:mi>e</m:mi>
								<m:mrow>
									<m:mi>k</m:mi>
									<m:mo>-</m:mo>
									<m:mn>1</m:mn>
								</m:mrow>
							</m:msub>
							<m:msub>
								<m:mi>v</m:mi>
								<m:mi>k</m:mi>
							</m:msub>
						</m:math>
					such that
					<m:math>
						<m:mo>∀</m:mo>
						<m:mi>i</m:mi>
						<m:mo>∈</m:mo>
						<m:mfenced close="]" open="[">
							<m:mn>1</m:mn>
							<m:mrow>
								<m:mi>k</m:mi>
								<m:mo>-</m:mo>
								<m:mn>1</m:mn>
							</m:mrow>
						</m:mfenced>
					</m:math>
					either
					<m:math>
						<m:msub>
							<m:mi>e</m:mi>
							<m:mi>i</m:mi>
						</m:msub>
						<m:mo>=</m:mo>
						<m:mfenced>
							<m:msub>
								<m:mi>v</m:mi>
								<m:mrow>
									<m:mi>i</m:mi>
									<m:mo>-</m:mo>
									<m:mn>1</m:mn>
								</m:mrow>
							</m:msub>
							<m:msub>
								<m:mi>v</m:mi>
								<m:mi>i</m:mi>
							</m:msub>
						</m:mfenced>
						<m:mo>∈</m:mo>
						<m:mi>E</m:mi>
					</m:math> 
					or 
					<m:math>
						<m:msub>
							<m:mi>e</m:mi>
							<m:mi>i</m:mi>
						</m:msub>
						<m:mo>=</m:mo>
						<m:mfenced>
							<m:msub>
								<m:mi>v</m:mi>
								<m:mi>i</m:mi>
							</m:msub>
							<m:mrow>
								<m:msub>
									<m:mi>v</m:mi>
									<m:mi>i</m:mi>
								</m:msub>
								<m:mo>-</m:mo>
								<m:mn>1</m:mn>
							</m:mrow>
						</m:mfenced>
						<m:mo>∈</m:mo>
						<m:mi>E</m:mi>
					</m:math>.
					</li><li>
					In directed graphs, a <i>directed walk</i> from vertex
					<m:math><m:mi>v</m:mi></m:math> to a vertex <m:math><m:mi>u</m:mi></m:math> is an
					alternating finite sequence of vertices and edges
						<m:math>
							<m:msub>
								<m:mi>v</m:mi>
								<m:mn>0</m:mn>
							</m:msub>
							<m:msub>
								<m:mi>e</m:mi>
								<m:mn>0</m:mn>
							</m:msub>
							<m:msub>
								<m:mi>v</m:mi>
								<m:mn>1</m:mn>
							</m:msub>
							<m:msub>
								<m:mi>e</m:mi>
								<m:mn>1</m:mn>
							</m:msub>
							<m:mo>…</m:mo>
							<m:msub>
								<m:mi>v</m:mi>
								<m:mrow>
									<m:mi>k</m:mi>
									<m:mo>-</m:mo>
									<m:mn>1</m:mn>
								</m:mrow>
							</m:msub>
							<m:msub>
								<m:mi>e</m:mi>
								<m:mrow>
									<m:mi>k</m:mi>
									<m:mo>-</m:mo>
									<m:mn>1</m:mn>
								</m:mrow>
							</m:msub>
							<m:msub>
								<m:mi>v</m:mi>
								<m:mi>k</m:mi>
							</m:msub>
						</m:math>
					such that
					<m:math>
						<m:mo>∀</m:mo>
						<m:mi>i</m:mi>
						<m:mo>∈</m:mo>
						<m:mfenced close="]" open="[">
							<m:mn>1</m:mn>
							<m:mrow>
								<m:mi>k</m:mi>
								<m:mo>-</m:mo>
								<m:mn>1</m:mn>
							</m:mrow>
						</m:mfenced>
						<m:mtext>,</m:mtext>
						<m:msub>
							<m:mi>e</m:mi>
							<m:mi>i</m:mi>
						</m:msub>
						<m:mo>=</m:mo>
						<m:mfenced>
							<m:msub>
								<m:mi>v</m:mi>
								<m:mrow>
									<m:mi>i</m:mi>
									<m:mo>-</m:mo>
									<m:mn>1</m:mn>
								</m:mrow>
							</m:msub>
							<m:msub>
								<m:mi>v</m:mi>
								<m:mi>i</m:mi>
							</m:msub>
						</m:mfenced>
						<m:mo>∈</m:mo>
						<m:mi>E</m:mi>
					</m:math>.
					</li><li>
					A <i>directed path</i> is a directed walk which does not
					repeat any vertex.
					</li><li>
					A <i>directed cycle</i> is a directed walk which does
					not repeat any vertex except for the first and the last.
					</li></ol>
				
				<div class="box">
					
					<p>
					A good glossary of graph-theoretical definitions can be found <a href="http://www.utm.edu/departments/math/graph/glossary.html">here</a>.
					</p>
					
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