GermanWikipedia - /page/entry.mu

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<title>Magischer Graph</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Magischer Graph</span></h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p><b>Magische Graphen</b> sind in der <a href="Graphentheorie" title="Graphentheorie">Graphentheorie</a> eine <a href="Graph_(Graphentheorie)" title="Graph (Graphentheorie)">Graphenklasse</a> mit speziellen Bewertungen von <a href="Knoten_(Graphentheorie)" title="Knoten (Graphentheorie)">Ecken</a> und <a href="Kante_(Graphentheorie)" title="Kante (Graphentheorie)">Kanten</a>. Das Gewicht einer Kante ist dabei gleich der Summe der Bewertungen der Anfangs-, Endecke und der Kante selbst. Sind alle Kantengewichte gleich, redet man von einem <b>kantenmagischen Graphen</b>. Das Gewicht einer Ecke ergibt sich als Summe der Eckenbewertung und der Bewertung jeder dort beginnenden Kante. Sind alle Eckengewichte gleich, so redet man von <b>eckenmagischen Graphen</b>. Graphen, die sowohl ecken- als auch kantenmagisch sind, werden <b>total magische Graphen</b> genannt.

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<div class="mw-heading mw-heading2"><h2 id="Eckenmagische_Graphen">Eckenmagische Graphen</h2></div>

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<annotation encoding="application/x-tex">{\displaystyle G=(E,K)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/378a300878ebae7f6933671fe142dfb0e5d6fbf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.61ex; height:2.843ex;" alt="{\displaystyle G=(E,K)}" loading="lazy"></span> ein <a href="Endlicher_Graph" class="mw-redirect" title="Endlicher Graph">endlicher</a> <a href="Einfacher_Graph" title="Einfacher Graph">einfacher</a> <a href="Ungerichteter_Graph" class="mw-redirect" title="Ungerichteter Graph">ungerichteter Graph</a> mit einer totalen Bewertung <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 \lambda :E\cup K\to \{1,2,...,|E|+|K|\}}">

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<annotation encoding="application/x-tex">{\displaystyle \lambda :E\cup K\to \{1,2,...,|E|+|K|\}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/67c169aee3d7c4f2ffd351ad99e83990a516c4ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.453ex; height:2.843ex;" alt="{\displaystyle \lambda :E\cup K\to \{1,2,...,|E|+|K|\}}" loading="lazy"></span>. <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}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span> bzw. <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 \lambda }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"></span> sind ecken-magisch, wenn

eine Eckenkonstante <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 h\in \mathbb {N} }">

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h\in \mathbb {N} }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd796ca77a508511c1ba760a6b508973b08afc36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.858ex; height:2.176ex;" alt="{\displaystyle h\in \mathbb {N} }" loading="lazy"></span> existiert, so dass für jede Ecke <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 e\in E}">

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<annotation encoding="application/x-tex">{\displaystyle e\in E}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34778736d9c6d607a4da3d25594b38dd3e8c82ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.7ex; height:2.176ex;" alt="{\displaystyle e\in E}" loading="lazy"></span> gilt:

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<annotation encoding="application/x-tex">{\displaystyle wt(e):=\lambda (e)+\sum _{eb\in K}\lambda (eb)=h}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c30f6aa07fa2efe1aa1e2680df7dc5b497ab9a1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.329ex; height:5.676ex;" alt="{\displaystyle wt(e):=\lambda (e)+\sum _{eb\in K}\lambda (eb)=h}" loading="lazy"></span> (Eckengewicht)</dd></dl>

<p>Gewicht einer Ecke ergibt sich als Summe der Eckenbewertung und der Bewertung jeder dort beginnenden Kante.

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<div class="mw-heading mw-heading2"><h2 id="Kantenmagische_Graphen">Kantenmagische Graphen</h2></div>

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<annotation encoding="application/x-tex">{\displaystyle G=(E,K)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/378a300878ebae7f6933671fe142dfb0e5d6fbf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.61ex; height:2.843ex;" alt="{\displaystyle G=(E,K)}" loading="lazy"></span> ein endlicher einfacher ungerichteter Graph mit einer totalen Bewertung <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 \lambda :E\cup K\to \{1,2,...,|E|+|K|\}}">

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</mrow>

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</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda :E\cup K\to \{1,2,...,|E|+|K|\}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/67c169aee3d7c4f2ffd351ad99e83990a516c4ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.453ex; height:2.843ex;" alt="{\displaystyle \lambda :E\cup K\to \{1,2,...,|E|+|K|\}}" loading="lazy"></span>. <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}">

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span> bzw. <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 \lambda }">

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"></span> sind kanten-magisch, wenn

eine Kantenkonstante <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 k\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {N} }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a5bc4b7383031ba693b7433198ead7170954c1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.73ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {N} }" loading="lazy"></span> existiert, so dass für jede Kante <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 ab\in K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ab\in K}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5994578e1de1db87278c8ff82007536356ee8c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.134ex; height:2.176ex;" alt="{\displaystyle ab\in K}" loading="lazy"></span> gilt:

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle wt(ab):=\lambda (a)+\lambda (ab)+\lambda (b)=k}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8c051e8491b6572e58bcc870bcc9b0edc3239320.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.225ex; height:2.843ex;" alt="{\displaystyle wt(ab):=\lambda (a)+\lambda (ab)+\lambda (b)=k}" loading="lazy"></span> (Kantengewicht)</dd></dl>

<p>Man gewichtet eine Kante mit der Summe der Bewertungen der Anfangs- und Endecke und der Kante selbst.

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<div class="mw-heading mw-heading2"><h2 id="Total_magische_Graphen">Total magische Graphen</h2></div>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle G=(E,K)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/378a300878ebae7f6933671fe142dfb0e5d6fbf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.61ex; height:2.843ex;" alt="{\displaystyle G=(E,K)}" loading="lazy"></span> ein endlicher einfacher ungerichteter Graph mit einer totalen Bewertung <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 \lambda :E\cup K\to \{1,2,...,|E|+|K|\}}">

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda :E\cup K\to \{1,2,...,|E|+|K|\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/67c169aee3d7c4f2ffd351ad99e83990a516c4ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.453ex; height:2.843ex;" alt="{\displaystyle \lambda :E\cup K\to \{1,2,...,|E|+|K|\}}" loading="lazy"></span>. <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}">

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span> bzw. <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 \lambda }">

</mstyle>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"></span> sind total magisch, wenn

eine Eckenkonstante <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 h\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h\in \mathbb {N} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd796ca77a508511c1ba760a6b508973b08afc36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.858ex; height:2.176ex;" alt="{\displaystyle h\in \mathbb {N} }" loading="lazy"></span> und eine Kantenkonstante <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 k\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {N} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a5bc4b7383031ba693b7433198ead7170954c1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.73ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {N} }" loading="lazy"></span> existiert, so dass <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span> bzw. <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 \lambda }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"></span> sowohl ecken- als auch kantenmagisch ist.

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

<ul><li>Der <a href="Vollst%C3%A4ndiger_Graph" title="Vollständiger Graph">triviale Graph</a> <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 K_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8520077dbcf03c2aabefd98d41a2269ed41a54fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.027ex; height:2.509ex;" alt="{\displaystyle K_{1}}" loading="lazy"></span> (Graph mit einer Ecke und keiner Kante) ist total magisch mit der Eckenkonstante <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 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"></span>. Die Kantenkonstante ist diskutabel.</li>

<li>Der <a href="Kreisgraph" title="Kreisgraph">Kreisgraph</a> <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 C_{3}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{3}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66e9abeb5057b7afbf88e3169101849354f13c65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\displaystyle C_{3}}" loading="lazy"></span> (Dreieck) ist total magisch.</li>

<li>Der <a href="Linearer_Graph" title="Linearer Graph">lineare Graph</a> <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 P_{3}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{3}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f92db8e65bb75b79799f0f3a29e975b37e227069.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{3}}" loading="lazy"></span> ist total magisch.</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{3}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f92db8e65bb75b79799f0f3a29e975b37e227069.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{3}}" loading="lazy"></span> und <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 K_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8520077dbcf03c2aabefd98d41a2269ed41a54fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.027ex; height:2.509ex;" alt="{\displaystyle K_{1}}" loading="lazy"></span> sind die einzigen total magischen <a href="Sterngraph" title="Sterngraph">Sterne</a>.</li>

<li>Der Graph <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 K_{1}\cup P_{3}}">

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<annotation encoding="application/x-tex">{\displaystyle K_{1}\cup P_{3}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a0b72744791746063c4d7e3ee6cc1476cbfafae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.156ex; height:2.509ex;" alt="{\displaystyle K_{1}\cup P_{3}}" loading="lazy"></span> ist total magisch.</li></ul>

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

<ul><li>Alison M. Marr, W. D. Wallis: <cite style="font-style:italic">Magic Graphs</cite>. 2. Auflage. Springer, 2012, ISBN 978-0-8176-8391-7.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Magischer+Graph&rft.au=Alison+M.+Marr%2C+W.+D.+Wallis&rft.btitle=Magic+Graphs&rft.date=2012&rft.edition=2.&rft.genre=book&rft.isbn=9780817683917&rft.pub=Springer" style="display:none"> </span></li>

<li>A. Kotzig, A. Rosa: <i>Magic valuations of finite graphs</i>. In: <i>Canad. Math. Bull.</i>, 13, 1970, S. 451–461</li></ul>

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

<ul><li><a href="Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/MagicGraph.html"><i>Magic graph</i>.</a> In: <i><a href="MathWorld" title="MathWorld">MathWorld</a></i> (englisch).</li>

<li><a rel="nofollow" class="external text" href="https://opensiuc.lib.siu.edu/cgi/viewcontent.cgi?referer=&httpsredir=1&article=1016&context=math_articles">Totally magic graphs</a></li></ul></div><div><div class="zim-footer">

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