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vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ádám's_conjecture"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Ádám's conjecture</span> </div> </a> <ul id="toc-Ádám's_conjecture-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algorithmic_questions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Algorithmic_questions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Algorithmic questions</span> </div> </a> <ul id="toc-Algorithmic_questions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" 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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Undirected graph acted on by a vertex-transitive cyclic group of symmetries</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For the square matrices, see <a href="/wiki/Circulant_matrix" title="Circulant matrix">Circulant matrix</a>.</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Paley13.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Paley13.svg/240px-Paley13.svg.png" decoding="async" width="240" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Paley13.svg/360px-Paley13.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/05/Paley13.svg/480px-Paley13.svg.png 2x" data-file-width="495" data-file-height="495" /></a><figcaption>The <a href="/wiki/Paley_graph" title="Paley graph">Paley graph</a> of order 13, an example of a circulant graph.</figcaption></figure> <p>In <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a>, a <b>circulant graph</b> is an <a href="/wiki/Undirected_graph" class="mw-redirect" title="Undirected graph">undirected graph</a> acted on by a <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic group</a> of <a href="/wiki/Graph_automorphism" title="Graph automorphism">symmetries</a> which <a href="/wiki/Vertex-transitive_graph" title="Vertex-transitive graph">takes any vertex to any other vertex</a>. It is sometimes called a <b>cyclic graph</b>,<sup id="cite_ref-ds1_1-0" class="reference"><a href="#cite_note-ds1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> but this term has other meanings. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Equivalent_definitions">Equivalent definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Circulant_graph&action=edit&section=1" title="Edit section: Equivalent definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Circulant graphs can be described in several equivalent ways:<sup id="cite_ref-v04_2-0" class="reference"><a href="#cite_note-v04-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <ul><li>The <a href="/wiki/Automorphism_group" title="Automorphism group">automorphism group</a> of the graph includes a <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic</a> <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> that <a href="/wiki/Group_action" title="Group action">acts transitively</a> on the graph's vertices. In other words, the graph has an <a href="/wiki/Automorphism" title="Automorphism">automorphism</a> which is a <a href="/wiki/Cyclic_permutation" title="Cyclic permutation">cyclic permutation</a> of its vertices.</li> <li>The graph has an <a href="/wiki/Adjacency_matrix" title="Adjacency matrix">adjacency matrix</a> that is a <a href="/wiki/Circulant_matrix" title="Circulant matrix">circulant matrix</a>.</li> <li>The <span class="texhtml mvar" style="font-style:italic;">n</span> vertices of the graph can be numbered from 0 to <span class="texhtml"><i>n</i> − 1</span> in such a way that, if some two vertices numbered <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml">(<i>x</i> + <i>d</i>) mod <i>n</i></span> are adjacent, then every two vertices numbered <span class="texhtml mvar" style="font-style:italic;">z</span> and <span class="texhtml">(<i>z</i> + <i>d</i>) mod <i>n</i></span> are adjacent.</li> <li>The graph can be drawn (possibly with crossings) so that its vertices lie on the corners of a <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygon</a>, and every rotational symmetry of the <a href="/wiki/Polygon" title="Polygon">polygon</a> is also a symmetry of the drawing.</li> <li>The graph is a <a href="/wiki/Cayley_graph" title="Cayley graph">Cayley graph</a> of a <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic group</a>.<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></li></ul> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Circulant_graph&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Every <a href="/wiki/Cycle_graph" title="Cycle graph">cycle graph</a> is a circulant graph, as is every <a href="/wiki/Crown_graph" title="Crown graph">crown graph</a> with number of vertices congruent to 2 modulo 4. </p><p>The <a href="/wiki/Paley_graph" title="Paley graph">Paley graphs</a> of order <span class="texhtml mvar" style="font-style:italic;">n</span> (where <span class="texhtml mvar" style="font-style:italic;">n</span> is a <a href="/wiki/Prime_number" title="Prime number">prime number</a> congruent to <span class="nowrap">1 modulo 4</span>) is a graph in which the vertices are the numbers from 0 to <span class="texhtml"><i>n</i> − 1</span> and two vertices are adjacent if their difference is a <a href="/wiki/Quadratic_residue" title="Quadratic residue">quadratic residue</a> modulo <span class="texhtml mvar" style="font-style:italic;">n</span>. Since the presence or absence of an edge depends only on the difference modulo <span class="texhtml mvar" style="font-style:italic;">n</span> of two vertex numbers, any Paley graph is a circulant graph. </p><p>Every <a href="/wiki/M%C3%B6bius_ladder" title="Möbius ladder">Möbius ladder</a> is a circulant graph, as is every <a href="/wiki/Complete_graph" title="Complete graph">complete graph</a>. A <a href="/wiki/Complete_bipartite_graph" title="Complete bipartite graph">complete bipartite graph</a> is a circulant graph if it has the same number of vertices on both sides of its bipartition. </p><p>If two numbers <span class="texhtml mvar" style="font-style:italic;">m</span> and <span class="texhtml mvar" style="font-style:italic;">n</span> are <a href="/wiki/Relatively_prime" class="mw-redirect" title="Relatively prime">relatively prime</a>, then the <span class="texhtml"><i>m</i> × <i>n</i></span> <a href="/wiki/Rook%27s_graph" title="Rook's graph">rook's graph</a> (a graph that has a vertex for each square of an <span class="texhtml"><i>m</i> × <i>n</i></span> <a href="/wiki/Chessboard" title="Chessboard">chessboard</a> and an edge for each two squares that a <a href="/wiki/Rook_(chess)" title="Rook (chess)">rook</a> can move between in a single move) is a circulant graph. This is because its symmetries include as a subgroup the cyclic group <i>C<sub>mn</sub></i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \simeq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≃</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \simeq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65b9738551241417d16d9843525ed52410af4dc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \simeq }"></span> <i>C<sub>m</sub></i>×<i>C<sub>n</sub></i>. More generally, in this case, the <a href="/wiki/Tensor_product_of_graphs" title="Tensor product of graphs">tensor product of graphs</a> between any <span class="texhtml mvar" style="font-style:italic;">m</span>- and <span class="texhtml mvar" style="font-style:italic;">n</span>-vertex circulants is itself a circulant.<sup id="cite_ref-v04_2-1" class="reference"><a href="#cite_note-v04-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Many of the known <a href="/wiki/Lower_bound" class="mw-redirect" title="Lower bound">lower bounds</a> on <a href="/wiki/Ramsey_number" class="mw-redirect" title="Ramsey number">Ramsey numbers</a> come from examples of circulant graphs that have small <a href="/wiki/Maximum_clique" class="mw-redirect" title="Maximum clique">maximum cliques</a> and small <a href="/wiki/Maximum_independent_set" class="mw-redirect" title="Maximum independent set">maximum independent sets</a>.<sup id="cite_ref-ds1_1-1" class="reference"><a href="#cite_note-ds1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="A_specific_example">A specific example</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Circulant_graph&action=edit&section=3" title="Edit section: A specific example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The circulant graph <span class="mwe-math-element"><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_{n}^{s_{1},\ldots ,s_{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{n}^{s_{1},\ldots ,s_{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c667f1890145f925f8294293b6ecfc81e5f1c3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.104ex; height:2.843ex;" alt="{\displaystyle C_{n}^{s_{1},\ldots ,s_{k}}}"></span> with jumps <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{1},\ldots ,s_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{1},\ldots ,s_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da75664c1fcca4186ddb1cac1218b1d87e8ae2e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.502ex; height:2.009ex;" alt="{\displaystyle s_{1},\ldots ,s_{k}}"></span> is defined as the graph with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> nodes labeled <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0,1,\ldots ,n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0,1,\ldots ,n-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebbc903de1e06d70a6b23d034fdad7333cd4c163" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.935ex; height:2.509ex;" alt="{\displaystyle 0,1,\ldots ,n-1}"></span> where each node <i>i</i> is adjacent to 2<i>k</i> nodes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\pm s_{1},\ldots ,i\pm s_{k}\mod n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>±</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>i</mi> <mo>±</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mspace width="1em" /> <mi>mod</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\pm s_{1},\ldots ,i\pm s_{k}\mod n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15fed35738e31b6cea992357a17b23ff81d3f394" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.444ex; height:2.509ex;" alt="{\displaystyle i\pm s_{1},\ldots ,i\pm s_{k}\mod n}"></span>. </p> <ul><li>The graph <span class="mwe-math-element"><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_{n}^{s_{1},\ldots ,s_{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{n}^{s_{1},\ldots ,s_{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c667f1890145f925f8294293b6ecfc81e5f1c3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.104ex; height:2.843ex;" alt="{\displaystyle C_{n}^{s_{1},\ldots ,s_{k}}}"></span> is <a href="/wiki/Connectivity_(graph_theory)" title="Connectivity (graph theory)">connected</a> if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gcd(n,s_{1},\ldots ,s_{k})=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gcd(n,s_{1},\ldots ,s_{k})=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba28036379c3939aae582608ad23e6be4c492e73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.488ex; height:2.843ex;" alt="{\displaystyle \gcd(n,s_{1},\ldots ,s_{k})=1}"></span>.</li> <li>If <span class="mwe-math-element"><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\leq s_{1}<\cdots <s_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mo>⋯</mo> <mo><</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq s_{1}<\cdots <s_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/101ad0c1792fd469d8e8647f0137f86e8eea1de6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.505ex; height:2.509ex;" alt="{\displaystyle 1\leq s_{1}<\cdots <s_{k}}"></span> are fixed <a href="/wiki/Integer" title="Integer">integers</a> then the number of <a href="/wiki/Spanning_tree" title="Spanning tree">spanning trees</a> <span class="mwe-math-element"><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(C_{n}^{s_{1},\ldots ,s_{k}})=na_{n}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">(</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t(C_{n}^{s_{1},\ldots ,s_{k}})=na_{n}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/450f1913e53730cf633eed143866d24aa05c6d06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.694ex; height:3.009ex;" alt="{\displaystyle t(C_{n}^{s_{1},\ldots ,s_{k}})=na_{n}^{2}}"></span> where <span class="mwe-math-element"><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_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"></span> satisfies a <a href="/wiki/Recurrence_relation" title="Recurrence relation">recurrence relation</a> of order <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{s_{k}-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{s_{k}-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73a1748ea98ce467afd51a9c77a37bd6e808c543" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.126ex; height:2.676ex;" alt="{\displaystyle 2^{s_{k}-1}}"></span>. <ul><li>In particular, <span class="mwe-math-element"><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(C_{n}^{1,2})=nF_{n}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">(</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t(C_{n}^{1,2})=nF_{n}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26f2a956935207735cfd505cd820ce198d4023ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.142ex; height:3.343ex;" alt="{\displaystyle t(C_{n}^{1,2})=nF_{n}^{2}}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"></span> is the <i>n</i>-th <a href="/wiki/Fibonacci_number" class="mw-redirect" title="Fibonacci number">Fibonacci number</a>.</li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="Self-complementary_circulants">Self-complementary circulants</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Circulant_graph&action=edit&section=4" title="Edit section: Self-complementary circulants"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Self-complementary_graph" title="Self-complementary graph">self-complementary graph</a> is a graph in which replacing every edge by a non-edge and vice versa produces an <a href="/wiki/Graph_isomorphism" title="Graph isomorphism">isomorphic graph</a>. For instance, a five-vertex cycle graph is self-complementary, and is also a circulant graph. More generally every <a href="/wiki/Paley_graph" title="Paley graph">Paley graph</a> of prime order is a self-complementary circulant graph.<sup id="cite_ref-s62_4-0" class="reference"><a href="#cite_note-s62-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Horst_Sachs" title="Horst Sachs">Horst Sachs</a> showed that, if a number <span class="texhtml mvar" style="font-style:italic;">n</span> has the property that every prime factor of <span class="texhtml mvar" style="font-style:italic;">n</span> is congruent to <span class="nowrap">1 modulo 4</span>, then there exists a self-complementary circulant with <span class="texhtml mvar" style="font-style:italic;">n</span> vertices. He <a href="/wiki/Conjecture" title="Conjecture">conjectured</a> that this condition is also necessary: that no other values of <span class="texhtml mvar" style="font-style:italic;">n</span> allow a self-complementary circulant to exist.<sup id="cite_ref-v04_2-2" class="reference"><a href="#cite_note-v04-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-s62_4-1" class="reference"><a href="#cite_note-s62-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The conjecture was <a href="/wiki/Mathematical_proof" title="Mathematical proof">proven</a> some 40 years later, by Vilfred.<sup id="cite_ref-v04_2-3" class="reference"><a href="#cite_note-v04-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Ádám's_conjecture"><span id=".C3.81d.C3.A1m.27s_conjecture"></span>Ádám's conjecture</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Circulant_graph&action=edit&section=5" title="Edit section: Ádám's conjecture"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Define a <i>circulant numbering</i> of a circulant graph to be a labeling of the vertices of the graph by the numbers from 0 to <span class="texhtml"><i>n</i> − 1</span> in such a way that, if some two vertices numbered <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span> are adjacent, then every two vertices numbered <span class="texhtml mvar" style="font-style:italic;">z</span> and <span class="texhtml">(<i>z</i> − <i>x</i> + <i>y</i>) mod <i>n</i></span> are adjacent. Equivalently, a circulant numbering is a numbering of the vertices for which the adjacency matrix of the graph is a circulant matrix. </p><p>Let <span class="texhtml mvar" style="font-style:italic;">a</span> be an integer that is relatively prime to <span class="texhtml mvar" style="font-style:italic;">n</span>, and let <span class="texhtml mvar" style="font-style:italic;">b</span> be any integer. Then the <a href="/wiki/Linear_function" title="Linear function">linear function</a> that takes a number <span class="texhtml mvar" style="font-style:italic;">x</span> to <span class="texhtml"><i>ax</i> + <i>b</i></span> transforms a circulant numbering to another circulant numbering. András Ádám conjectured that these linear maps are the only ways of renumbering a circulant graph while preserving the circulant property: that is, if <span class="texhtml mvar" style="font-style:italic;">G</span> and <span class="texhtml mvar" style="font-style:italic;">H</span> are isomorphic circulant graphs, with different numberings, then there is a linear map that transforms the numbering for <span class="texhtml mvar" style="font-style:italic;">G</span> into the numbering for <span class="texhtml mvar" style="font-style:italic;">H</span>. However, Ádám's conjecture is now known to be false. A <a href="/wiki/Counterexample" title="Counterexample">counterexample</a> is given by graphs <span class="texhtml mvar" style="font-style:italic;">G</span> and <span class="texhtml mvar" style="font-style:italic;">H</span> with 16 vertices each; a vertex <span class="texhtml mvar" style="font-style:italic;">x</span> in <span class="texhtml mvar" style="font-style:italic;">G</span> is connected to the six neighbors <span class="texhtml"><i>x</i> ± 1</span>, <span class="texhtml"><i>x</i> ± 2</span>, and <span class="texhtml"><i>x</i> ± 7</span> modulo 16, while in <span class="texhtml mvar" style="font-style:italic;">H</span> the six neighbors are <span class="texhtml"><i>x</i> ± 2</span>, <span class="texhtml"><i>x</i> ± 3</span>, and <span class="texhtml"><i>x</i> ± 5</span> modulo 16. These two graphs are isomorphic, but their isomorphism cannot be realized by a linear map.<sup id="cite_ref-v04_2-4" class="reference"><a href="#cite_note-v04-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Toida%27s_conjecture" title="Toida's conjecture">Toida's conjecture</a> refines Ádám's conjecture by considering only a special class of circulant graphs, in which all of the differences between adjacent graph vertices are relatively prime to the number of vertices. According to this refined conjecture, these special circulant graphs should have the property that all of their symmetries come from symmetries of the underlying additive group of numbers modulo <span class="texhtml"><i>n</i></span>. It was proven by two groups in 2001 and 2002.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Algorithmic_questions">Algorithmic questions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Circulant_graph&action=edit&section=6" title="Edit section: Algorithmic questions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There is a <a href="/wiki/Polynomial-time" class="mw-redirect" title="Polynomial-time">polynomial-time</a> recognition algorithm for circulant graphs, and the isomorphism problem for circulant graphs can be solved in polynomial time.<sup id="cite_ref-muz04_7-0" class="reference"><a href="#cite_note-muz04-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ep04_8-0" class="reference"><a href="#cite_note-ep04-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Circulant_graph&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-ds1-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-ds1_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ds1_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/DS1">Small Ramsey Numbers</a>, Stanisław P. Radziszowski, <i><a href="/wiki/Electronic_Journal_of_Combinatorics" title="Electronic Journal of Combinatorics">Electronic J. Combinatorics</a></i>, dynamic survey 1, updated 2014.</span> </li> <li id="cite_note-v04-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-v04_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-v04_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-v04_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-v04_2-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-v04_2-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFVilfred2004" class="citation cs2">Vilfred, V. (2004), "On circulant graphs", in Balakrishnan, R.; Sethuraman, G.; Wilson, Robin J. 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Sci., vol. 56, Providence, Rhode Island: American Mathematical Society, pp. 241–264, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1816402">1816402</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+isomorphism+problem+for+circulant+graphs+via+Schur+ring+theory&rft.btitle=Codes+and+association+schemes+%28Piscataway%2C+NJ%2C+1999%29&rft.place=Providence%2C+Rhode+Island&rft.series=DIMACS+Ser.+Discrete+Math.+Theoret.+Comput.+Sci.&rft.pages=241-264&rft.pub=American+Mathematical+Society&rft.date=2001&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1816402%23id-name%3DMR&rft.aulast=Muzychuk&rft.aufirst=Mikhail&rft.au=Klin%2C+Mikhail&rft.au=P%C3%B6schel%2C+Reinhard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACirculant+graph" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDobsonMorris2002" class="citation cs2">Dobson, Edward; <a href="/wiki/Joy_Morris" title="Joy Morris">Morris, Joy</a> (2002), <a rel="nofollow" class="external text" href="https://www.combinatorics.org/Volume_9/Abstracts/v9i1r35.html">"Toida's conjecture is true"</a>, <i>Electronic Journal of Combinatorics</i>, <b>9</b> (1): R35:1–R35:14, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1928787">1928787</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Electronic+Journal+of+Combinatorics&rft.atitle=Toida%27s+conjecture+is+true&rft.volume=9&rft.issue=1&rft.pages=R35%3A1-R35%3A14&rft.date=2002&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1928787%23id-name%3DMR&rft.aulast=Dobson&rft.aufirst=Edward&rft.au=Morris%2C+Joy&rft_id=https%3A%2F%2Fwww.combinatorics.org%2FVolume_9%2FAbstracts%2Fv9i1r35.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACirculant+graph" class="Z3988"></span></span> </li> <li id="cite_note-muz04-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-muz04_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMuzychuk2004" class="citation journal cs1">Muzychuk, Mikhail (2004). "A Solution of the Isomorphism Problem for Circulant Graphs". <i>Proc. London Math. Soc</i>. <b>88</b>: 1–41. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fs0024611503014412">10.1112/s0024611503014412</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2018956">2018956</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proc.+London+Math.+Soc.&rft.atitle=A+Solution+of+the+Isomorphism+Problem+for+Circulant+Graphs&rft.volume=88&rft.pages=1-41&rft.date=2004&rft_id=info%3Adoi%2F10.1112%2Fs0024611503014412&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2018956%23id-name%3DMR&rft.aulast=Muzychuk&rft.aufirst=Mikhail&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACirculant+graph" class="Z3988"></span></span> </li> <li id="cite_note-ep04-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-ep04_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEvdokimovPonomarenko2004" class="citation journal cs1">Evdokimov, Sergei; Ponomarenko, Ilia (2004). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs1061-0022-04-00833-7">"Recognition and verification of an isomorphism of circulant graphs in polynomial time"</a>. <i>St. Petersburg Math. J</i>. <b>15</b>: 813–835. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs1061-0022-04-00833-7">10.1090/s1061-0022-04-00833-7</a></span>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2044629">2044629</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=St.+Petersburg+Math.+J.&rft.atitle=Recognition+and+verification+of+an+isomorphism+of+circulant+graphs+in+polynomial+time&rft.volume=15&rft.pages=813-835&rft.date=2004&rft_id=info%3Adoi%2F10.1090%2Fs1061-0022-04-00833-7&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2044629%23id-name%3DMR&rft.aulast=Evdokimov&rft.aufirst=Sergei&rft.au=Ponomarenko%2C+Ilia&rft_id=https%3A%2F%2Fdoi.org%2F10.1090%252Fs1061-0022-04-00833-7&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACirculant+graph" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Circulant_graph&action=edit&section=8" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Circulant_Graph"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/CirculantGraph.html">"Circulant Graph"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Circulant+Graph&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FCirculantGraph.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACirculant+graph" class="Z3988"></span></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Circulant_graph&oldid=1240296209">https://en.wikipedia.org/w/index.php?title=Circulant_graph&oldid=1240296209</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Graph_families" title="Category:Graph families">Graph families</a></li><li><a href="/wiki/Category:Regular_graphs" title="Category:Regular graphs">Regular graphs</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 14 August 2024, at 17:01<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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