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<!doctype html> <html lang="en" class="discretemathematics historyandterminology mathworldcontributors"> <head> <title>Line Graph -- from Wolfram MathWorld</title> <meta name="DC.Title" content="Line Graph" /> <meta name="DC.Creator" content="Weisstein, Eric W." /> <meta name="DC.Description" content="A line graph L(G) (also called an adjoint, conjugate, covering, derivative, derived, edge, edge-to-vertex dual, interchange, representative, or theta-obrazom graph) of a simple graph G is obtained by associating a vertex with each edge of the graph and connecting two vertices with an edge iff the corresponding edges of G have a vertex in common (Gross and Yellen 2006, p. 20). Given a line graph L(G), the underlying graph G is known as the root graph of L(G). The root graph of a simple line..." /> <meta name="description" content="A line graph L(G) (also called an adjoint, conjugate, covering, derivative, derived, edge, edge-to-vertex dual, interchange, representative, or theta-obrazom graph) of a simple graph G is obtained by associating a vertex with each edge of the graph and connecting two vertices with an edge iff the corresponding edges of G have a vertex in common (Gross and Yellen 2006, p. 20). Given a line graph L(G), the underlying graph G is known as the root graph of L(G). The root graph of a simple line..." /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2001-12-23" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2005-09-09" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2006-01-29" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2006-11-08" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2007-05-09" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2009-04-08" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2010-11-12" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2011-12-10" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2012-08-15" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2013-03-01" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2013-08-02" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2014-09-05" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2015-11-03" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2023-03-09" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2023-05-10" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Discrete Mathematics:Graph Theory:Simple Graphs:Line Graphs" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Discrete Mathematics:Graph Theory:Graph Operations" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Discrete Mathematics:Graph Theory:Forbidden Induced Subgraphs" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:History and Terminology:Wolfram Language Commands" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:MathWorld Contributors:Naverniouk" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:MathWorld Contributors:Pegg" /> <meta name="DC.Subject" scheme="MSC_2000" content="05C" /> <meta name="DC.Rights" content="Copyright 1999-2024 Wolfram Research, Inc. 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Given a line graph L(G), the underlying graph G is known as the root graph of L(G). The root graph of a simple line..."> <meta name="twitter:card" content="summary_large_image"> <meta name="twitter:site" content="@WolframResearch"> <meta name="twitter:title" content="Line Graph -- from Wolfram MathWorld"> <meta name="twitter:description" content="A line graph L(G) (also called an adjoint, conjugate, covering, derivative, derived, edge, edge-to-vertex dual, interchange, representative, or theta-obrazom graph) of a simple graph G is obtained by associating a vertex with each edge of the graph and connecting two vertices with an edge iff the corresponding edges of G have a vertex in common (Gross and Yellen 2006, p. 20). Given a line graph L(G), the underlying graph G is known as the root graph of L(G). 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</li> <li> <a href="/topics/GraphTheory.html">Graph Theory</a> </li> <li> <a href="/topics/GraphOperations.html">Graph Operations</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/DiscreteMathematics.html">Discrete Mathematics</a> </li> <li> <a href="/topics/GraphTheory.html">Graph Theory</a> </li> <li> <a href="/topics/ForbiddenInducedSubgraphs.html">Forbidden Induced Subgraphs</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/HistoryandTerminology.html">History and Terminology</a> </li> <li> <a href="/topics/WolframLanguageCommands.html">Wolfram Language Commands</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/MathWorldContributors.html">MathWorld Contributors</a> </li> <li> <a href="/topics/Naverniouk.html">Naverniouk</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/MathWorldContributors.html">MathWorld Contributors</a> </li> <li> <a href="/topics/Pegg.html">Pegg</a> </li> </ul><a class="show-more">More...</a><a class="display-n show-less">Less...</a></nav>   <h1>Line Graph</h1>  <hr class="margin-t-1-8 margin-b-3-4">  <div class="attachments text-align-r"> <a href="/notebooks/GraphTheory/LineGraph.nb" download="LineGraph.nb"><img src="/images/entries/download-notebook-icon.png" width="26" height="27" alt="DOWNLOAD Mathematica Notebook" /><span>Download <span class="display-i display-n__600">Wolfram </span>Notebook</span></a> </div>  <div class="entry-content"> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="358.413" height="230.028" src="images/eps-svg/LineGraph_800.svg" class="" alt="LineGraph" /> </div> <p> A line graph <img src="/images/equations/LineGraph/Inline1.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="38" height="21" alt="L(G)" /> (also called an adjoint, conjugate, covering, derivative, derived, edge, edge-to-vertex dual, interchange, representative, or <img src="/images/equations/LineGraph/Inline2.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="8" height="21" alt="theta" />-obrazom graph) of a <a href="/SimpleGraph.html">simple graph</a> <img src="/images/equations/LineGraph/Inline3.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> is obtained by associating a vertex with each edge of the graph and connecting two vertices with an edge <a href="/Iff.html">iff</a> the corresponding edges of <img src="/images/equations/LineGraph/Inline4.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> have a vertex in common (Gross and Yellen 2006, p. 20). </p> <p> Given a line graph <img src="/images/equations/LineGraph/Inline5.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="38" height="21" alt="L(G)" />, the underlying graph <img src="/images/equations/LineGraph/Inline6.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> is known as the <a href="/RootGraph.html">root graph</a> of <img src="/images/equations/LineGraph/Inline7.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="38" height="21" alt="L(G)" />. The <a href="/RootGraph.html">root graph</a> of a <a href="/SimpleGraph.html">simple</a> line graph is unique, except for the case <img src="/images/equations/LineGraph/Inline8.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="60" height="22" alt="K_3=C_3" /> (Harary 1994, pp. 72-73). </p> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="348.244" height="216.219" src="images/eps-svg/LineGraphDirected_800.svg" class="" alt="LineGraphDirected" /> </div> <p> The line graph of a <a href="/DirectedGraph.html">directed graph</a> <img src="/images/equations/LineGraph/Inline9.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> is the <a href="/DirectedGraph.html">directed graph</a> <img src="/images/equations/LineGraph/Inline10.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="38" height="21" alt="L(G)" /> whose <a href="/VertexSet.html">vertex set</a> corresponds to the <a href="/ArcSet.html">arc set</a> of <img src="/images/equations/LineGraph/Inline11.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> and having an <a href="/GraphArc.html">arc</a> directed from an edge <img src="/images/equations/LineGraph/Inline12.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="15" height="21" alt="e_1" /> to an edge <img src="/images/equations/LineGraph/Inline13.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="15" height="21" alt="e_2" /> if in <img src="/images/equations/LineGraph/Inline14.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" />, the head of <img src="/images/equations/LineGraph/Inline15.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="15" height="21" alt="e_1" /> meets the tail of <img src="/images/equations/LineGraph/Inline16.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="15" height="21" alt="e_2" /> (Gross and Yellen 2006, p. 265). </p> <p> Line graphs are implemented in the <a href="http://www.wolfram.com/language/">Wolfram Language</a> as <tt><a href="http://reference.wolfram.com/language/ref/LineGraph.html">LineGraph</a></tt>[<i>g</i>]. Precomputed line graph identifications of many named graphs can be obtained in the <a href="http://www.wolfram.com/language/">Wolfram Language</a> using <tt><a href="http://reference.wolfram.com/language/ref/GraphData.html">GraphData</a></tt>[<i>graph</i>, <tt>"LineGraphName"</tt>]. </p> <p> The numbers of simple line graphs on <img src="/images/equations/LineGraph/Inline17.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="39" height="21" alt="n=1" />, 2, ... vertices are 1, 2, 4, 10, 24, 63, 166, 471, 1408, ... (OEIS <a href="http://oeis.org/A132220">A132220</a>), and the numbers of connected simple line graphs are 1, 1, 2, 5, 12, 30, 79, 227, ... (OEIS <a href="http://oeis.org/A003089">A003089</a>). </p> <p> The following table summarizes some named graphs and their corresponding line graphs. </p> <div class="table-responsive"> <table align="center" class="mathworldtable"> <tr style=""><td align="left"><img src="/images/equations/LineGraph/Inline18.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /></td><td align="left"><img src="/images/equations/LineGraph/Inline19.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="38" height="21" alt="L(G)" /></td></tr><tr style=""><td align="left"><a href="/ClawGraph.html">claw graph</a> <img src="/images/equations/LineGraph/Inline20.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="23" alt="K_(1,3)" /></td><td align="left"><a href="/TriangleGraph.html">triangle graph</a> <img src="/images/equations/LineGraph/Inline21.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="22" alt="C_3" /></td></tr><tr style=""><td align="left"><a href="/CompleteBipartiteGraph.html">complete bipartite graph</a> <img src="/images/equations/LineGraph/Inline22.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="23" alt="K_(2,3)" /></td><td align="left"><a href="/PrismGraph.html">prism graph</a> <img src="/images/equations/LineGraph/Inline23.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="17" height="22" alt="Y_3" /></td></tr><tr style=""><td align="left"><a href="/CubicalGraph.html">cubical graph</a></td><td align="left"><a href="/CuboctahedralGraph.html">cuboctahedral graph</a></td></tr><tr style=""><td align="left"><a href="/CycleGraph.html">cycle graph</a> <img src="/images/equations/LineGraph/Inline24.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="21" alt="C_n" /></td><td align="left"><a href="/CycleGraph.html">cycle graph</a> <img src="/images/equations/LineGraph/Inline25.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="21" alt="C_n" /></td></tr><tr style=""><td align="left"><a href="/DodecahedralGraph.html">dodecahedral graph</a></td><td align="left"><a href="/IcosidodecahedralGraph.html">icosidodecahedral graph</a></td></tr><tr style=""><td align="left"><a href="/PathGraph.html">path graph</a> <img src="/images/equations/LineGraph/Inline26.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="21" alt="P_2" /></td><td align="left"><a href="/SingletonGraph.html">singleton graph</a> <img src="/images/equations/LineGraph/Inline27.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="20" height="21" alt="K_1" /></td></tr><tr style=""><td align="left"><a href="/PathGraph.html">path graph</a> <img src="/images/equations/LineGraph/Inline28.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="21" alt="P_n" /></td><td align="left"><a href="/PathGraph.html">path graph</a> <img src="/images/equations/LineGraph/Inline29.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="33" height="22" alt="P_(n-1)" /></td></tr><tr style=""><td align="left"><a href="/SquareGraph.html">square graph</a> <img src="/images/equations/LineGraph/Inline30.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="21" alt="C_4" /></td><td align="left"><a href="/SquareGraph.html">square graph</a> <img src="/images/equations/LineGraph/Inline31.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="21" alt="C_4" /></td></tr><tr style=""><td align="left"><a href="/StarGraph.html">star graph</a> <img src="/images/equations/LineGraph/Inline32.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="16" height="22" alt="S_5" /></td><td align="left"><a href="/TetrahedralGraph.html">tetrahedral graph</a> <img src="/images/equations/LineGraph/Inline33.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="20" height="21" alt="K_4" /></td></tr><tr style=""><td align="left"><a href="/StarGraph.html">star graph</a> <img src="/images/equations/LineGraph/Inline34.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="16" height="22" alt="S_6" /></td><td align="left"><a href="/Pentatope.html">pentatope</a> graph <img src="/images/equations/LineGraph/Inline35.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="20" height="22" alt="K_5" /></td></tr><tr style=""><td align="left"><a href="/StarGraph.html">star graph</a> <img src="/images/equations/LineGraph/Inline36.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="16" height="21" alt="S_n" /></td><td align="left"><a href="/CompleteGraph.html">complete graph</a> <img src="/images/equations/LineGraph/Inline37.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="34" height="22" alt="K_(n-1)" /></td></tr><tr style=""><td align="left"><a href="/TetrahedralGraph.html">tetrahedral graph</a> <img src="/images/equations/LineGraph/Inline38.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="20" height="21" alt="K_4" /></td><td align="left"><a href="/OctahedralGraph.html">octahedral graph</a></td></tr><tr style=""><td align="left"><a href="/TriangleGraph.html">triangle graph</a> <img src="/images/equations/LineGraph/Inline39.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="22" alt="C_3" /></td><td align="left"><a href="/TriangleGraph.html">triangle graph</a> <img src="/images/equations/LineGraph/Inline40.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="22" alt="C_3" /></td></tr> </table> </div> <p> Line graphs are <a href="/Claw-FreeGraph.html">claw-free</a>. </p> <p> The line graph of a <a href="/Graph.html">graph</a> with <img src="/images/equations/LineGraph/Inline41.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> nodes, <img src="/images/equations/LineGraph/Inline42.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="8" height="21" alt="e" /> edges, and vertex degrees <img src="/images/equations/LineGraph/Inline43.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="14" height="21" alt="d_i" /> contains <img src="/images/equations/LineGraph/Inline44.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="43" height="21" alt="n^'=e" /> nodes and </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/LineGraph/NumberedEquation1.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="111" height="50" alt=" e^'=1/2sum_(i=1)^nd_i^2-e " /></td></tr> </table> </div> <p> edges (Skiena 1990, p. 137). The <a href="/IncidenceMatrix.html">incidence matrix</a> <img src="/images/equations/LineGraph/Inline45.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="23" alt="C" /> of a graph and <a href="/AdjacencyMatrix.html">adjacency matrix</a> <img src="/images/equations/LineGraph/Inline46.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="10" height="23" alt="L" /> of its line graph are related by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/LineGraph/NumberedEquation2.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="102" height="20" alt=" L=C^(T)C-2I, " /></td></tr> </table> </div> <p> where <img src="/images/equations/LineGraph/Inline47.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="5" height="23" alt="I" /> is the <a href="/IdentityMatrix.html">identity matrix</a> (Skiena 1990, p. 136). </p> <p> Krausz (1943) proved that a solution <img src="/images/equations/LineGraph/Inline48.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="74" height="21" alt="L(H)=G" /> exists for a simple graph <img src="/images/equations/LineGraph/Inline49.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> <a href="/Iff.html">iff</a> <img src="/images/equations/LineGraph/Inline50.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> decomposes into complete subgraphs with each vertex of <img src="/images/equations/LineGraph/Inline51.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> appearing in at most two members of the decomposition. This theorem, however, is not useful for implementation of an efficient algorithm because of the possibly large number of decompositions involved (West 2000, p. 280). </p> <p> van Rooij and Wilf (1965) shows that a solution to <img src="/images/equations/LineGraph/Inline52.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="74" height="21" alt="L(H)=G" /> exists for a simple graph <img src="/images/equations/LineGraph/Inline53.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> <a href="/Iff.html">iff</a> <img src="/images/equations/LineGraph/Inline54.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> is <a href="/Claw-FreeGraph.html">claw-free</a> and no induced <a href="/DiamondGraph.html">diamond graph</a> of <img src="/images/equations/LineGraph/Inline55.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> has two odd triangles. Here, a triangular subgraph <img src="/images/equations/LineGraph/Inline56.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="21" alt="T" /> is said to be even if the neighborhood <img src="/images/equations/LineGraph/Inline57.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="36" height="21" alt="N(v)" /> and vertex set <img src="/images/equations/LineGraph/Inline58.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="39" height="21" alt="V(T)" /> intersect in an odd number of points for some <img src="/images/equations/LineGraph/Inline59.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="67" height="21" alt="v in V(G)" /> and even if <img src="/images/equations/LineGraph/Inline60.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="41" height="21" alt="N(V)" /> and <img src="/images/equations/LineGraph/Inline61.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="39" height="21" alt="V(T)" /> intersect in an even number of points for every <img src="/images/equations/LineGraph/Inline62.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="67" height="21" alt="v in V(G)" /> (West 2000, p. 281). </p> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="405.76" height="334.752" src="images/eps-svg/NonLineGraphs_800.svg" class="" alt="NonLineGraphs" /> </div> <p> A <a href="/SimpleGraph.html">simple graph</a> is a line graph of some <a href="/SimpleGraph.html">simple graph</a> <a href="/Iff.html">iff</a> if does not contain any of the above nine <a href="/BeinekeGraphs.html">Beineke graphs</a> as a <a href="/ForbiddenInducedSubgraph.html">forbidden induced subgraph</a> (van Rooij and Wilf 1965; Beineke 1968; Skiena 1990, p. 138; Harary 1994, pp. 74-75; West 2000, p. 282; Gross and Yellen 2006, p. 405). This statement is sometimes known as the Beineke theorem. These nine graphs are implemented in the <a href="http://www.wolfram.com/language/">Wolfram Language</a> as <tt><a href="http://reference.wolfram.com/language/ref/GraphData.html">GraphData</a></tt>[<tt>"Beineke"</tt>]. Of the nine, one has four nodes (the <a href="/ClawGraph.html">claw graph</a> = <a href="/StarGraph.html">star graph</a> <img src="/images/equations/LineGraph/Inline63.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="16" height="21" alt="S_4" /> = <a href="/CompleteBipartiteGraph.html">complete bipartite graph</a> <img src="/images/equations/LineGraph/Inline64.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="23" alt="K_(1,3)" />), two have five nodes, and six have six nodes (including the <a href="/WheelGraph.html">wheel graph</a> <img src="/images/equations/LineGraph/Inline65.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="22" height="22" alt="W_6" />). </p> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="456.48" height="247.26" src="images/eps-svg/NonLineGraphs5_1000.svg" class="" alt="NonLineGraphs5" /> </div> <p> A graph with <a href="/MinimumVertexDegree.html">minimum vertex degree</a> at least 5 is a line graph <a href="/Iff.html">iff</a> it does not contain any of the above six <a href="/MetelskyGraphs.html">Metelsky graphs</a> as an <a href="/InducedSubgraph.html">induced subgraph</a> (Metelsky and Tyshkevich 1997). These six graphs are implemented in the <a href="http://www.wolfram.com/language/">Wolfram Language</a> as <tt><a href="http://reference.wolfram.com/language/ref/GraphData.html">GraphData</a></tt>[<tt>"Metelsky"</tt>]. </p> <p> A graph is not a line graph if the smallest element of its <a href="/GraphSpectrum.html">graph spectrum</a> is less than <img src="/images/equations/LineGraph/Inline66.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="21" height="21" alt="-2" /> (Van Mieghem, 2010, Liu <i>et al. </i>2010). </p> <p> Whitney (1932) showed that, with the exception of <img src="/images/equations/LineGraph/Inline67.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="20" height="22" alt="K_3" /> and <img src="/images/equations/LineGraph/Inline68.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="23" alt="K_(1,3)" />, any two <a href="/ConnectedGraph.html">connected graphs</a> with isomorphic line graphs are isomorphic (Skiena 1990, p. 138). </p> <p> Lehot (1974) gave a linear time algorithm that reconstructs the original graph from its line graph. Liu <i>et al. </i>(2010) give an <img src="/images/equations/LineGraph/Inline69.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="43" height="25" alt="O(n^2)" /> algorithm for reconstructing the original graph from its line graph, where <img src="/images/equations/LineGraph/Inline70.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> is the number of vertices in the line graph. This algorithm is more time efficient than the efficient algorithm of Roussopoulos (1973). </p> <p> The line graph of an <a href="/EulerianGraph.html">Eulerian graph</a> is both Eulerian and <a href="/HamiltonianGraph.html">Hamiltonian</a> (Skiena 1990, p. 138). More information about cycles of line graphs is given by Harary and Nash-Williams (1965) and Chartrand (1968). </p> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="491.35" height="277.375" src="images/eps-svg/LineGraphs_1000.svg" class="" alt="LineGraphs" /> </div> <p> Taking the line graph twice does not return the original graph unless the line graph of a graph <img src="/images/equations/LineGraph/Inline71.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> is isomorphic to <img src="/images/equations/LineGraph/Inline72.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> itself. In fact, The only <a href="/ConnectedGraph.html">connected graph</a> that is isomorphic to its line graph is a <a href="/CycleGraph.html">cycle graph</a> <img src="/images/equations/LineGraph/Inline73.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="21" alt="C_n" /> for <img src="/images/equations/LineGraph/Inline74.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="n>=3" /> (Skiena 1990, p. 137). Graph unions of cycle graphs (e.g., <img src="/images/equations/LineGraph/Inline75.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="32" height="22" alt="2C_3" />, <img src="/images/equations/LineGraph/Inline76.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="57" height="22" alt="C_3+C_4" />, etc.) are also isomorphic to their line graphs, so the graphs that are isomorphic to their line graphs are the regular graphs of degree 2, and the total numbers of not-necessarily connected simple graphs that are isomorphic to their lines graphs are given by the number of partitions of their <a href="/VertexCount.html">vertex count</a> having smallest part <img src="/images/equations/LineGraph/Inline77.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="28" height="21" alt=">=3" />, given for <img src="/images/equations/LineGraph/Inline78.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="39" height="21" alt="n=1" />, 2, ... by 0, 0, 1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 10, 13, 17, ... (OEIS <a href="http://oeis.org/A026796">A026796</a>), the first few of which are illustrated above. </p> </div>  <hr class="margin-b-1-1-4"> <div class="c-777 entry-secondary-content">  <h2>See also</h2><a href="/BeinekeGraphs.html">Beineke Graphs</a>, <a href="/Claw-FreeGraph.html">Claw-Free Graph</a>, <a href="/MetelskyGraphs.html">Metelsky Graphs</a>, <a href="/RootGraph.html">Root Graph</a>, <a href="/TotalGraph.html">Total Graph</a>       <h2>Explore with Wolfram|Alpha</h2> <div id="WAwidget"> <div class="WAwidget-wrapper"> <img alt="WolframAlpha" title="WolframAlpha" src="/images/wolframalpha/WA-logo.png" width="136" height="20"> <form name="wolframalpha" action="https://www.wolframalpha.com/input/" target="_blank"> <input type="text" name="i" class="search" placeholder="Solve your math problems and get step-by-step solutions" value=""> <button type="submit" title="Evaluate on WolframAlpha"></button> </form> </div> <div class="WAwidget-wrapper try"> <p class="text-align-r"> More things to try: </p> <ul> <li> <a target="_blank" href="http://www.wolframalpha.com/input/?i=line+graph"> line graph </a> </li> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=Bode+plot+of+s%2F%281-s%29+sampling+period+.02">Bode plot of s/(1-s) sampling period .02</a></li> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=focal+parameter+of+an+ellipse+with+semiaxes+4%2C3">focal parameter of an ellipse with semiaxes 4,3</a></li> </ul> </div> </div>   <h2>References</h2><cite>Beineke, L. W. "Derived Graphs and Digraphs." In <i>Beiträge zur Graphentheorie</i> (Ed. H. Sachs, H. Voss, and H. Walther). Leipzig, Germany: Teubner, pp. 17-33, 1968.</cite><cite>Beineke, L. W. "Characterizations of Derived Graphs." <i>J. Combin. Th.</i> <b>9</b>, 129-135, 1970.</cite><cite>Chartrand, G. "On Hamiltonian Line Graphs." <i>Trans. Amer. Math. Soc.</i> <b>134</b>, 559-566, 1968.</cite><cite>Degiorgi, D. G. and Simon, K. "A Dynamic Algorithm for Line Graph Recognition." In <i>WG '95: Proceedings of the 21st International Workshop on Graph-Theoretic Concepts in Computer Science.</i> London: Springer-Verlag, pp. 37-48, 1995.</cite><cite>DistanceRegular.org. "Line Graphs." <a href="http://www.distanceregular.org/indexes/linegraphs.html">http://www.distanceregular.org/indexes/linegraphs.html</a>.</cite><cite>Gross, J. T. and Yellen, J. <i><a href="http://www.amazon.com/exec/obidos/ASIN/158488505X/ref=nosim/ericstreasuretro">Graph Theory and Its Applications, 2nd ed.</a></i> Boca Raton, FL: CRC Press, pp. 20 and 265, 2006.</cite><cite>Harary, F. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0201410338/ref=nosim/ericstreasuretro">Graph Theory.</a></i> Reading, MA: Addison-Wesley, 1994.</cite><cite>Harary, F. and Nash-Williams, C. J. A. "On Eulerian and Hamiltonian Graphs and Line Graphs." <i>Canad. Math. Bull.</i> <b>8</b>, 701-709, 1965.</cite><cite>Krausz, J. "Démonstration nouvelle d'une théorème de Whitney sur les réseaux." <i>Mat. Fiz. Lapok</i> <b>50</b>, 78-89, 1943.</cite><cite>Lehot, P. G. H. "An Optimal Algorithm to Detect a Line Graph and Output Its Root Graph." <i>J. ACM</i> <b>21</b>, 569-575, 1974.</cite><cite>Liu, D.; Trajanovski, S.; and Van Mieghem, P. "Reverse Line Graph Construction: The Matrix Relabeling Algorithm MARINLINGA Versus Roussopoulos's Algorithm." 22 Oct 2010. <a href="http://arxiv.org/abs/1005.0943">http://arxiv.org/abs/1005.0943</a>.</cite><cite>Metelsky, Yu. and Tyshkevich, R. "On Line Graphs of Linear 3-Uniform Hypergraphs." <i>J. Graph Th.</i> <b>25</b>, 243-251, 1997.</cite><cite>Naor, J. and Novick, M. B. "An Efficient Reconstruction of a Graph from Its Line Graph in Parallel." <i>J. Algorithms</i> <b>11</b>, 132-143, 1990.</cite><cite>Roussopoulos, N. D. "A <img src="/images/equations/LineGraph/Inline79.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="79" height="21" alt="max{m,n}" /> Algorithm for Determining the Graph <img src="/images/equations/LineGraph/Inline80.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="15" height="21" alt="H" /> From Its Line Graph <img src="/images/equations/LineGraph/Inline81.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" />." <i>Info. Proc. Let.</i> <b>2</b>, 108-112, 1973.</cite><cite>Saaty, T. L. and Kainen, P. C. "Line Graphs." §4-3 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/0486650928/ref=nosim/ericstreasuretro">The Four-Color Problem: Assaults and Conquest.</a></i> New York: Dover, pp. 108-112, 1986.</cite><cite>Skiena, S. "Line Graph." §4.1.5 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/0521806860/ref=nosim/ericstreasuretro">Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica.</a></i> Reading, MA: Addison-Wesley, pp. 128 and 135-139, 1990.</cite><cite>Sloane, N. J. A. Sequences <a href="http://oeis.org/A003089">A003089</a>/M1417, <a href="http://oeis.org/A026796">A026796</a>, and <a href="http://oeis.org/A132220">A132220</a> in "The On-Line Encyclopedia of Integer Sequences."</cite><cite>&Scaron;oltés, Ľ. "Forbidden Induced Subgraphs for Line Graphs." <i>Disc. Math.</i> <b>132</b>, 391-394, 1994.</cite><cite>Van Mieghem, P. <i><a href="http://www.amazon.com/exec/obidos/ASIN/052119458X/ref=nosim/ericstreasuretro">Graph Spectra for Complex Networks.</a></i> Cambridge, England: Cambridge University Press, 2010.</cite><cite>van Rooij, A. and Wilf, H. "The Interchange Graph of a Finite Graph." <i>Acta Math. Acad. Sci. Hungar.</i> <b>16</b>, 263-269, 1965.</cite><cite>Roussopoulos, N. D. "A <img src="/images/equations/LineGraph/Inline82.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="79" height="21" alt="max{m,n}" /> Algorithm for Determining the Graph <img src="/images/equations/LineGraph/Inline83.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="h" /> from its Line Graph <img src="/images/equations/LineGraph/Inline84.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="10" height="21" alt="g" />." <i>Inform. Proc. Lett.</i> <b>2</b>, 108-112, 1973.</cite><cite>West, D. B. "Characterizing Line Graphs." <i><a href="http://www.amazon.com/exec/obidos/ASIN/0130144002/ref=nosim/ericstreasuretro">Introduction to Graph Theory, 2nd ed.</a></i> Englewood Cliffs, NJ: Prentice-Hall, pp. 279-282, 2000.</cite><cite>Whitney, H. "Congruent Graphs and the Connectivity of Graphs." <i>Amer. J. Math.</i> <b>54</b>, 150-168, 1932.</cite><h2>Referenced on Wolfram|Alpha</h2><a href="http://www.wolframalpha.com/entities/mathworld/line_graph/03/nr/aw/" title="Line Graph" target="_blank">Line Graph</a>   <h2>Cite this as:</h2> <p> <a href="/about/author.html">Weisstein, Eric W.</a> "Line Graph." From <a href="/"><i>MathWorld</i></a>--A Wolfram Web Resource. <a href="https://mathworld.wolfram.com/LineGraph.html">https://mathworld.wolfram.com/LineGraph.html</a> </p>  <h2>Subject classifications</h2><nav class="breadcrumbs"><ul class="breadcrumb"> <li> <a href="/topics/DiscreteMathematics.html">Discrete Mathematics</a> </li> <li> <a href="/topics/GraphTheory.html">Graph Theory</a> </li> <li> <a href="/topics/SimpleGraphs.html">Simple Graphs</a> </li> <li> <a href="/topics/LineGraphs.html">Line Graphs</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/DiscreteMathematics.html">Discrete Mathematics</a> </li> <li> <a href="/topics/GraphTheory.html">Graph Theory</a> </li> <li> <a href="/topics/GraphOperations.html">Graph Operations</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/DiscreteMathematics.html">Discrete Mathematics</a> </li> <li> <a href="/topics/GraphTheory.html">Graph Theory</a> </li> <li> <a href="/topics/ForbiddenInducedSubgraphs.html">Forbidden Induced Subgraphs</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/HistoryandTerminology.html">History and Terminology</a> </li> <li> <a href="/topics/WolframLanguageCommands.html">Wolfram Language Commands</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/MathWorldContributors.html">MathWorld Contributors</a> </li> <li> <a href="/topics/Naverniouk.html">Naverniouk</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/MathWorldContributors.html">MathWorld Contributors</a> </li> <li> <a href="/topics/Pegg.html">Pegg</a> </li> </ul><a class="show-more">More...</a><a class="display-n show-less">Less...</a></nav>  </div> </section> </section>  </div> </main> <aside id="bottom"> <style> #bottom { padding-bottom: 65px; } #acknowledgment { display:none; } .attribution { font-size: .75rem; font-style: italic; } footer ul li:not(:last-of-type)::after { background: #a3a3a3; margin-left: .3rem; margin-right: .1rem; } @media all and (max-width: 900px) { .attribution { font-size: 12px; } } @media (max-width: 600px) { footer { max-width: 360px; } footer ul { max-width: 360px; } footer ul:nth-child(1) li:nth-child(2):after { content: ""; height: 11px; } footer ul:nth-child(1) li:nth-child(3):after { content: ""; height: 0px; } } </style> <footer> <ul> <li><a href="/about/">About MathWorld</a></li> <li><a href="/classroom/">MathWorld Classroom</a></li> <li><a href="/contact/">Contribute</a></li> <li><a href="https://www.amazon.com/exec/obidos/ASIN/1420072218/ref=nosim/weisstein-20" target="_blank">MathWorld Book</a></li> <li class="display-n display-ib__600"><a href="https://www.wolfram.com" target="_blank">wolfram.com</a></li> </ul> <ul> <li class="display-n__600"><a href="/whatsnew/">13,208 Entries</a></li> <li class="display-n__600"><a href="/whatsnew/">Last Updated: Thu Nov 21 2024</a></li>  <li><a href="https://www.wolfram.com" target="_blank">©1999–2024 Wolfram Research, Inc.</a></li> <li><a href="https://www.wolfram.com/legal/terms/mathworld.html" target="_blank">Terms of Use</a></li> </ul> <ul class="wolfram"> <li class="display-n__600 display-n__900"><a href="https://www.wolfram.com" target="_blank" aria-label="Wolfram"><img src="/images/footer/wolfram-logo.png" alt="Wolfram" title="Wolfram" width="121" height="28"></a></li> <li class="display-n__600"><a href="https://www.wolfram.com" target="_blank">wolfram.com</a></li> <li class="display-n__600"><a href="https://www.wolfram.com/education/" target="_blank">Wolfram for Education</a></li> <li class="attribution">Created, developed and nurtured by Eric Weisstein at Wolfram Research</li> </ul> </footer> <section id="acknowledgment"> <i>Created, developed and nurtured by Eric Weisstein at Wolfram Research</i> </section> </aside> <script type="text/javascript" src="/scripts/scripts.js"></script> <script src="/common/js/c2c/1.0/WolframC2C.js"></script> <script src="/common/js/c2c/1.0/WolframC2CGui.js"></script> <script src="/common/js/c2c/1.0/WolframC2CDefault.js"></script> <link rel="stylesheet" href="/common/js/c2c/1.0/WolframC2CGui.css.en"> <style> .wolfram-c2c-wrapper { padding: 0px !important; border: 0px; } .wolfram-c2c-wrapper:active { border: 0px; } .wolfram-c2c-wrapper:hover { border: 0px; } </style> <script> let c2cWrittings = new WolframC2CDefault({'triggerClass':'mathworld-c2c_above', 'uniqueIdPrefix': 'mathworld-c2c_above-'}); </script> <style> #IPstripe-outer { background: #47a2af; } #IPstripe-outer:hover { background: #0095aa; } </style> <div id="IPstripe-wrap"></div> <script src="/common/stripe/stripe.en.js"></script> </body> </html>