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<title>Multigraph</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Multigraph</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about the mathematical concept. For other uses, see <a href="Multigraph_(disambiguation)" class="mw-disambig" title="Multigraph (disambiguation)">Multigraph (disambiguation)</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">"Pseudograph" redirects here; not to be confused with <a href="Pseudepigraph" class="mw-redirect" title="Pseudepigraph">Pseudepigraph</a>.</div>

<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, and more specifically in <a href="Graph_theory" title="Graph theory">graph theory</a>, a <b>multigraph</b> is a <a href="Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a> which is permitted to have <a href="Multiple_edges" title="Multiple edges">multiple edges</a> (also called <i>parallel edges</i><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>), that is, <a href="Glossary_of_graph_theory#Basics" title="Glossary of graph theory">edges</a> that have the same <a href="Vertex_(graph_theory)" title="Vertex (graph theory)">end nodes</a>. Thus two vertices may be connected by more than one edge.

</p><p>There are 2 distinct notions of multiple edges:

</p>

<ul><li><i>Edges without own identity</i>: The identity of an edge is defined solely by the two nodes it connects. In this case, the term "multiple edges" means that the same edge can occur several times between these two nodes.</li>

<li><i>Edges with own identity</i>: Edges are primitive entities just like nodes. When multiple edges connect two nodes, these are different edges.</li></ul>

<p>A multigraph is different from a <a href="Hypergraph" title="Hypergraph">hypergraph</a>, which is a graph in which an edge can connect any number of nodes, not just two.

</p><p>For some authors, the terms <i>pseudograph</i> and <i>multigraph</i> are synonymous. For others, a <b>pseudograph</b> is a multigraph that is permitted to have <a href="Loop_(graph_theory)" title="Loop (graph theory)">loops</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Undirected_multigraph_(edges_without_own_identity)">Undirected multigraph (edges without own identity)</h2></div>

<p>A multigraph <i>G</i> is an <a href="Ordered_pair" title="Ordered pair">ordered pair</a> <i>G</i> := (<i>V</i>, <i>E</i>) with

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<ul><li><i>V</i> a <a href="Set_(mathematics)" title="Set (mathematics)">set</a> of <i>vertices</i> or <i>nodes</i>,</li>

<li><i>E</i> a <a href="Multiset" title="Multiset">multiset</a> of unordered pairs of vertices, called <i>edges</i> or <i>lines</i>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Undirected_multigraph_(edges_with_own_identity)">Undirected multigraph (edges with own identity)</h2></div>

<p>A multigraph <i>G</i> is an ordered <a href="Triple_(mathematics)" class="mw-redirect" title="Triple (mathematics)">triple</a> <i>G</i> := (<i>V</i>, <i>E</i>, <i>r</i>) with

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<ul><li><i>V</i> a <a href="Set_(mathematics)" title="Set (mathematics)">set</a> of <i>vertices</i> or <i>nodes</i>,</li>

<li><i>E</i> a <a href="Set_(mathematics)" title="Set (mathematics)">set</a> of <i>edges</i> or <i>lines</i>,</li>

<li><i>r</i> : <i>E</i> → {{<i>x</i>,<i>y</i>} : <i>x</i>, <i>y</i> ∈ <i>V</i>}, assigning to each edge an unordered pair of endpoint nodes.</li></ul>

<p>Some authors allow multigraphs to have <a href="Loop_(graph_theory)" title="Loop (graph theory)">loops</a>, that is, an edge that connects a vertex to itself,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> while others call these <b>pseudographs</b>, reserving the term multigraph for the case with no loops.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Directed_multigraph_(edges_without_own_identity)">Directed multigraph (edges without own identity)</h2></div>

<p>A <b>multidigraph</b> is a <a href="Directed_graph" title="Directed graph">directed graph</a> which is permitted to have <i>multiple arcs,</i> i.e., arcs with the same source and target nodes. A multidigraph <i>G</i> is an ordered pair <i>G</i> := (<i>V</i>, <i>A</i>) with

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<ul><li><i>V</i> a set of <i>vertices</i> or <i>nodes</i>,</li>

<li><i>A</i> a multiset of ordered pairs of vertices called <i>directed edges</i>, <i>arcs</i> or <i>arrows</i>.</li></ul>

<p>A <b>mixed multigraph</b> <i>G</i> := (<i>V</i>, <i>E</i>, <i>A</i>) may be defined in the same way as a <a href="Mixed_graph" title="Mixed graph">mixed graph</a>.

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<div class="mw-heading mw-heading2"><h2 id="Directed_multigraph_(edges_with_own_identity)">Directed multigraph (edges with own identity)</h2></div>

<p>A multidigraph or <a href="Quiver_(mathematics)" title="Quiver (mathematics)">quiver</a> <i>G</i> is an ordered <a href="Tuple" title="Tuple">4-tuple</a> <i>G</i> := (<i>V</i>, <i>A</i>, <i>s</i>, <i>t</i>) with

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<ul><li><i>V</i> a <a href="Set_(mathematics)" title="Set (mathematics)">set</a> of <i>vertices</i> or <i>nodes</i>,</li>

<li><i>A</i> a <a href="Set_(mathematics)" title="Set (mathematics)">set</a> of <i>edges</i> or <i>lines</i>,</li>

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<annotation encoding="application/x-tex">{\displaystyle s:A\rightarrow V}</annotation>

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</math></span><img src="./22cf89d5943d2ede4dce63ee8c8e610bf4b91d4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.172ex; height:2.176ex;" alt="{\displaystyle s:A\rightarrow V}" loading="lazy"></span>, assigning to each edge its source node,</li>

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<annotation encoding="application/x-tex">{\displaystyle t:A\rightarrow V}</annotation>

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</math></span><img src="./67f2d34d9fab03234e7730ec04fb8e5b2a67baab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.921ex; height:2.176ex;" alt="{\displaystyle t:A\rightarrow V}" loading="lazy"></span>, assigning to each edge its target node.</li></ul>

<p>This notion might be used to model the possible flight connections offered by an airline. In this case the multigraph would be a <a href="Directed_graph" title="Directed graph">directed graph</a> with pairs of directed parallel edges connecting cities to show that it is possible to fly both <i>to</i> and <i>from</i> these locations.

</p><p>In <a href="Category_theory" title="Category theory">category theory</a> a small <a href="Category_(mathematics)" title="Category (mathematics)">category</a> can be defined as a multidigraph (with edges having their own identity) equipped with an associative composition law and a distinguished self-loop at each vertex serving as the left and right identity for composition. For this reason, in category theory the term <i>graph</i> is standardly taken to mean "multidigraph", and the underlying multidigraph of a category is called its <b>underlying digraph</b>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Labeling">Labeling</h2></div>

<p>Multigraphs and multidigraphs also support the notion of <a href="Graph_labeling" title="Graph labeling">graph labeling</a>, in a similar way. However there is no unity in terminology in this case.

</p><p>The definitions of <b>labeled multigraphs</b> and <b>labeled multidigraphs</b> are similar, and we define only the latter ones here.

</p><p><i>Definition 1</i>: A labeled multidigraph is a <a href="Labeled_graph" class="mw-redirect" title="Labeled graph">labeled graph</a> with <i>labeled</i> arcs.

</p><p>Formally: A labeled multidigraph G is a multigraph with <i>labeled</i> vertices and arcs. Formally it is an 8-tuple <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=(\Sigma _{V},\Sigma _{A},V,A,s,t,\ell _{V},\ell _{A})}">

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<annotation encoding="application/x-tex">{\displaystyle G=(\Sigma _{V},\Sigma _{A},V,A,s,t,\ell _{V},\ell _{A})}</annotation>

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</math></span><img src="./3757cbd5ddd5a4472afe39c5448de9d32180d6d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.65ex; height:2.843ex;" alt="{\displaystyle G=(\Sigma _{V},\Sigma _{A},V,A,s,t,\ell _{V},\ell _{A})}" loading="lazy"></span> where

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> is a set of vertices and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> is a set of arcs.</li>

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<annotation encoding="application/x-tex">{\displaystyle \Sigma _{V}}</annotation>

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</math></span><img src="./fb662593b0fed0c5f952393646de91e446a22e29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.174ex; height:2.509ex;" alt="{\displaystyle \Sigma _{V}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{A}}">

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<annotation encoding="application/x-tex">{\displaystyle \Sigma _{A}}</annotation>

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</math></span><img src="./1c8f81a07827960795eac0a4d3f1a9fd7774d1c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle \Sigma _{A}}" loading="lazy"></span> are finite alphabets of the available vertex and arc labels,</li>

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<annotation encoding="application/x-tex">{\displaystyle s\colon A\rightarrow \ V}</annotation>

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</math></span><img src="./38dac4b8641632725146bebd76a4c594dcfe470b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.849ex; height:2.176ex;" alt="{\displaystyle s\colon A\rightarrow \ V}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\colon A\rightarrow \ V}">

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<annotation encoding="application/x-tex">{\displaystyle t\colon A\rightarrow \ V}</annotation>

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</math></span><img src="./0e45fca09e1f9f41d6f8b26989b6f58c987ac220.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.599ex; height:2.176ex;" alt="{\displaystyle t\colon A\rightarrow \ V}" loading="lazy"></span> are two maps indicating the <i>source</i> and <i>target</i> vertex of an arc,</li>

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<annotation encoding="application/x-tex">{\displaystyle \ell _{V}\colon V\rightarrow \Sigma _{V}}</annotation>

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</math></span><img src="./b862999b6edb76f3e65825af4bffd781930c8bce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.075ex; height:2.509ex;" alt="{\displaystyle \ell _{V}\colon V\rightarrow \Sigma _{V}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell _{A}\colon A\rightarrow \Sigma _{A}}">

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<annotation encoding="application/x-tex">{\displaystyle \ell _{A}\colon A\rightarrow \Sigma _{A}}</annotation>

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</math></span><img src="./d43a3a998d0d07b2636497634ff87c0b896cc942.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.969ex; height:2.509ex;" alt="{\displaystyle \ell _{A}\colon A\rightarrow \Sigma _{A}}" loading="lazy"></span> are two maps describing the labeling of the vertices and arcs.</li></ul>

<p><i>Definition 2</i>: A labeled multidigraph is a <a href="Labeled_graph" class="mw-redirect" title="Labeled graph">labeled graph</a> with multiple <i>labeled</i> arcs, i.e. arcs with the same end vertices and the same arc label (note that this notion of a labeled graph is different from the notion given by the article <a href="Graph_labeling" title="Graph labeling">graph labeling</a>).

</p>

<ul><li><a href="Multidimensional_network" title="Multidimensional network">Multidimensional network</a></li>

<li><a href="Glossary_of_graph_theory_terms" class="mw-redirect" title="Glossary of graph theory terms">Glossary of graph theory terms</a></li>

<li><a href="Graph_theory" title="Graph theory">Graph theory</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">For example, see Balakrishnan 1997, p. 1 or Chartrand and Zhang 2012, p. 26.</span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">For example, see Bollobás 2002, p. 7 or Diestel 2010, p. 28.</span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">For example, see Wilson 2002, p. 6 or Chartrand and Zhang 2012, pp. 26-27.</span>

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFBalakrishnan1997" class="citation book cs1">Balakrishnan, V. K. (1997). <i>Graph Theory</i>. McGraw-Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-005489-4</bdi>.</cite></li>

<li><cite id="CITEREFBollobás2002" class="citation book cs1"><a href="B%C3%A9la_Bollob%C3%A1s" title="Béla Bollobás">Bollobás, Béla</a> (2002). <i>Modern Graph Theory</i>. <a href="Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>. Vol. 184. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-98488-7</bdi>.</cite></li>

<li><cite id="CITEREFChartrandZhang2012" class="citation book cs1"><a href="Gary_Chartrand" title="Gary Chartrand">Chartrand, Gary</a>; <a href="Ping_Zhang_(graph_theorist)" title="Ping Zhang (graph theorist)">Zhang, Ping</a> (2012). <i>A First Course in Graph Theory</i>. Dover. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-48368-9</bdi>.</cite></li>

<li><cite id="CITEREFDiestel2010" class="citation book cs1">Diestel, Reinhard (2010). <i>Graph Theory</i>. Graduate Texts in Mathematics. Vol. 173 (4th ed.). Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-642-14278-9</bdi>.</cite></li>

<li><cite id="CITEREFGrossYellen1998" class="citation book cs1">Gross, Jonathan L.; Yellen, Jay (1998). <i>Graph Theory and Its Applications</i>. CRC Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8493-3982-0</bdi>.</cite></li>

<li><cite id="CITEREFGrossYellen2003" class="citation book cs1">Gross, Jonathan L.; Yellen, Jay, eds. (2003). <i>Handbook of Graph Theory</i>. CRC. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-58488-090-2</bdi>.</cite></li>

<li><cite id="CITEREFHarary1995" class="citation book cs1"><a href="Frank_Harary" title="Frank Harary">Harary, Frank</a> (1995). <i>Graph Theory</i>. Addison Wesley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-201-41033-8</bdi>.</cite></li>

<li><cite id="CITEREFJansonKnuthLuczakPittel1993" class="citation journal cs1"><a href="Svante_Janson" title="Svante Janson">Janson, Svante</a>; <a href="Donald_Knuth" title="Donald Knuth">Knuth, Donald E.</a>; Luczak, Tomasz; Pittel, Boris (1993). "The birth of the giant component". <i>Random Structures and Algorithms</i>. <b>4</b> (3): <span class="nowrap">231–</span>358. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/9310236">math/9310236</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1993math.....10236J">1993math.....10236J</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Frsa.3240040303">10.1002/rsa.3240040303</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1042-9832">1042-9832</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1220220">1220220</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:206454812">206454812</a>.</cite></li>

<li><cite id="CITEREFWilson2002" class="citation book cs1"><a href="Robert_A._Wilson_(mathematician)" class="mw-redirect" title="Robert A. Wilson (mathematician)">Wilson, Robert A.</a> (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=iq0sSnIxJioC&dq=pseudograph&pg=PA6"><i>Graphs, Colourings and the Four-Colour Theorem</i></a>. Oxford Science Publ. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-19-851062-4</bdi>.</cite></li>

<li><cite id="CITEREFZwillinger2002" class="citation book cs1">Zwillinger, Daniel (2002). <i>CRC Standard Mathematical Tables and Formulae</i> (31st ed.). Chapman & Hall/CRC. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-58488-291-3</bdi>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><span class="noviewer" typeof="mw:File"><span></span></span> This article incorporates <a href="Copyright_status_of_works_by_the_federal_government_of_the_United_States" title="Copyright status of works by the federal government of the United States">public domain material</a> from <cite id="CITEREFPaul_E._Black" class="citation cs1">Paul E. Black. <a rel="nofollow" class="external text" href="https://xlinux.nist.gov/dads/HTML/multigraph.html">"Multigraph"</a>. <i><a href="Dictionary_of_Algorithms_and_Data_Structures" class="mw-redirect" title="Dictionary of Algorithms and Data Structures">Dictionary of Algorithms and Data Structures</a></i>. <a href="NIST" class="mw-redirect" title="NIST">NIST</a>.</cite></li></ul></div><div><div class="zim-footer">

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