<?xml version="1.0" encoding="UTF-8"?>
<?xml-stylesheet type="text/xsl" href="../../xslt/mathml.xsl"?><html xmlns="http://www.w3.org/1999/xhtml" xmlns:m="http://www.w3.org/1998/Math/MathML">
  <head>
    <meta http-equiv="Content-Type" content="text/html; charset=utf-8"/>
    <title>Graph Theory: Industrial Drilling</title>
    <link rel="stylesheet" href="../../css/style_mozilla.css" type="text/css" media="screen"/>
  </head>
  <body>
    <div id="logobarleft">
      <a href="http://links.math.rpi.edu/">
        <img src="../../css/titlebar_left.gif" alt="Project Links Logo"/>
      </a>
    </div>
    <div id="logobarright">
      <img src="../../css/titlebar_right.jpg" alt="Artwork"/>
    </div>
    <div id="titlebar">Graph Theory: Industrial Drilling</div>
    <div id="breadcrumbbar">
			Home &gt; Graph Theory: Industrial Drilling
					&gt; Foundations
					&gt; Edges and Vertices</div>
    <div id="sidebar">
      <div class="toolmenu1">
        <a href="banana.html" class="toolmenu1">Crib Sheet</a>
      </div>
      <br/>
      <div class="toolmenu2">
        <a href="banana.html" class="toolmenu2">Library</a>
      </div>
      <br/>
      <div class="toolmenu3">
        <a href="banana.html" class="toolmenu3">Help</a>
      </div>
      <br/>
      <div class="toolmenu4">
        <a href="banana.html" class="toolmenu4">Map</a>
      </div>
      <br/>
      <div class="toolmenu5">
        <a href="problemlist.xml" class="toolmenu5">Problem List</a>
      </div>
      <br/>
      <div class="toolmenu5">
        <a href="../index.html" class="toolmenu5">Module Home</a>
      </div>
      <br/>
      <div class="toolmenu6">
        <a href="http://links.math.rpi.edu/" class="toolmenu6">Links Home</a>
      </div>
      <br/>
      <br/>
      <div class="toc_topic">
        <a class="toc_link" href="page1.xml">The Drilling Problem</a>
      </div>
      <div class="toc_topic">
        <a class="toc_link" href="page4.xml">Objectives</a>
      </div>
      <div class="toc_topic">
        <a class="toc_link" href="">Foundations</a>
      </div>
      <div class="toc_subtopic">
        <a class="toc_link" href="page5.xml">What is a Graph?</a>
      </div>
      <div class="toc_subtopic">
        <a class="toc_link" href="page7.xml">Edges and Vertices</a>
      </div>
      <div class="toc_subtopic">
        <a class="toc_link" href="page10.xml">A First Theorem</a>
      </div>
      <div class="toc_subtopic">
        <a class="toc_link" href="page12.xml">Types of Graphs</a>
      </div>
      <div class="toc_subtopic">
        <a class="toc_link" href="page15.xml">Subgraphs and Supergraphs</a>
      </div>
      <div class="toc_subtopic">
        <a class="toc_link" href="page18.xml">Walks, Paths, Cycles</a>
      </div>
      <div class="toc_subtopic">
        <a class="toc_link" href="page21.xml">Trees</a>
      </div>
      <div class="toc_topic">
        <a class="toc_link" href="">Solving the Drilling Problem</a>
      </div>
      <div class="toc_subtopic">
        <a class="toc_link" href="page24.xml">Independent Sets, Cliques, etc.</a>
      </div>
      <div class="toc_subtopic">
        <a class="toc_link" href="page26.xml">Vertex Coloring</a>
      </div>
      <div class="toc_subtopic">
        <a class="toc_link" href="page28.xml">Modeling it as a Graph Theory Problem</a>
      </div>
      <div class="toc_subtopic">
        <a class="toc_link" href="page31.xml">First Lower Bound</a>
      </div>
      <div class="toc_subtopic">
        <a class="toc_link" href="page33.xml">The Inside View</a>
      </div>
      <div class="toc_subtopic">
        <a class="toc_link" href="page34.xml">A Good Coloring</a>
      </div>
      <div class="toc_subtopic">
        <a class="toc_link" href="page37.xml">The Best Coloring</a>
      </div>
      <div class="toc_topic">
        <a class="toc_link" href="page38.xml">What's Next?</a>
      </div>
    </div>
    <div id="sectionbar">
      <div id="spacerright" class="spacerright">-</div>
      <div>
        <a href="page7.xml" class="sectionshighlighted">concepts</a>
      </div>
      <div>
        <a href="" class="sectionsgrayed">discover</a>
      </div>
      <div>
        <a href="" class="sectionsgrayed">apply</a>
      </div>
      <div>
        <a href="page8.xml" class="sections">collaborate</a>
      </div>
      <div>
        <a href="page9.xml" class="sections">practice</a>
      </div>
      <div id="spacerleft" class="spacerleft">-</div>
    </div>
    <div id="content"><h1>Edges and Vertices</h1>
				<p>
					Now that you understand what a graph is, let's define some
					useful terms relative to the edges and vertices of a graph:
				</p>
				<p>
					If an edge
					<m:math>
						<m:mi>e</m:mi>
						<m:mo>=</m:mo>
						<m:mfenced>
							<m:mi>a</m:mi>
							<m:mi>b</m:mi>
						</m:mfenced>
					</m:math> is an edge in a graph, then
				</p>
				<ol><li>
						<m:math><m:mi>e</m:mi></m:math> is said to be
						<i>incident</i> with vertices
						<m:math><m:mi>a</m:mi></m:math> and
						<m:math><m:mi>b</m:mi></m:math>.
					</li><li>
						<m:math><m:mi>a</m:mi></m:math> and <m:math><m:mi>b</m:mi></m:math> are also said to be
						<i>incident</i> with <m:math><m:mi>e</m:mi></m:math>.
						</li><li>
						<m:math><m:mi>a</m:mi></m:math> and <m:math><m:mi>b</m:mi></m:math> are
						<i>adjacent</i> to one another.
					</li></ol>
				<p>
					The <i>degree</i> of a vertex
					<m:math><m:mi>v</m:mi></m:math> is the number of vertices
					adjacent to <m:math><m:mi>v</m:mi></m:math>.  (This equals
					the number of edges incident with
					<m:math><m:mi>v</m:mi></m:math>).
				</p>
				<div class="box">
					<b><i>Example:</i></b>
					
						<div class="figure" align="center"><a name=""/><img src="../graphics/simplegraph1.gif" alt="Degree"/><br/><b><i>Figure 5:
		Degree</i></b><br/><i>
								In this graph, vertices
								<m:math><m:mi>a</m:mi></m:math>,
								<m:math><m:mi>b</m:mi></m:math>, and
								<m:math><m:mi>c</m:mi></m:math> have degree 2, while
								vertex <m:math><m:mi>d</m:mi></m:math> is an
								<i>isolated</i> vertex (its degree is 0).
								Notation:
								<m:math>
									<m:msub>
										<m:mi>deg</m:mi>
										<m:mi>G</m:mi>
									</m:msub>
									<m:mfenced>
										<m:mi>v</m:mi>
									</m:mfenced>
									<m:mo>=</m:mo>
									<m:mi>deg</m:mi>
									<m:mfenced>
										<m:mi>v</m:mi>
									</m:mfenced>
								</m:math>.
							</i><br/><br/></div>
					
				</div>
				<p>
					An <i>isolated</i> vertex is a vertex of degree
					<m:math><m:mn>0</m:mn></m:math>.  In the example graph
					above, <m:math><m:mi>d</m:mi></m:math> is an isolated
					vertex.
				</p>
				<p>
					A graph is <i>regular</i> if the degrees of all
					vertices are equal.  A graph is
					<i>r-regular</i> if
					<m:math>
						<m:mo>∀</m:mo>
						<m:mi>x</m:mi>
						<m:mo>∈</m:mo>
						<m:mi>V</m:mi>
						<m:mfenced>
							<m:mi>G</m:mi>
						</m:mfenced>
					</m:math>,
					<m:math>
						<m:msub>
							<m:mi>deg</m:mi>
							<m:mi>G</m:mi>
						</m:msub>
						<m:mfenced>
							<m:mi>v</m:mi>
						</m:mfenced>
						<m:mo>=</m:mo>
						<m:mi>r</m:mi>
					</m:math>.
				</p>
				<h2>When are graphs the same?</h2>
				<p>
					A graph is fully defined by its vertex and edge sets.
					However, the same graph can be drawn differently.  Different
					ways to represent the same graph are often considered
					immaterial.  This approach is captured by the definition of
					<i>isomorphic</i> graphs:
				</p>
				<div class="box">
					<b><i>Isomorphism</i></b>
					
						<p>
							Two graphs
							<m:math>
								<m:mi>G</m:mi>
								<m:mo>=</m:mo>
								<m:mfenced>
									<m:mi>V</m:mi>
									<m:mi>E</m:mi>
								</m:mfenced>
							</m:math> and
							<m:math>
								<m:mi>H</m:mi>
								<m:mo>=</m:mo>
								<m:mfenced>
									<m:mi>U</m:mi>
									<m:mi>F</m:mi>
								</m:mfenced>
							</m:math> are called <i>isomorphic</i> if there is
							a one-to-one correspondence between
							<m:math><m:mi>V</m:mi></m:math> and
							<m:math><m:mi>U</m:mi></m:math> which preserves all
							adjacencies.
						</p>
					
				</div>
			</div>
    <div id="nav">
      <a href="page6.xml" class="nav">back</a>
      <a href="page8.xml" class="nav">next</a>
    </div>
    <div class="copyright">
			Copyright 1999
			Rensselaer Polytechnic Institute. All Rights Reserved.
		</div>
  </body>
</html>
