Distance modulus - Revision history

en>Monkbot: /* Usage */Fix CS1 format parameter errors

2014-04-06T15:03:19Z

← Older revision		Revision as of 17:03, 6 April 2014
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en>Jwagnerhki: distance is luminosity distance

2014-02-07T21:36:32Z

distance is luminosity distance

← Older revision		Revision as of 23:36, 7 February 2014
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	~~{{infobox graph~~		The author is recognized by the name of Numbers Lint. It's not a typical thing but what she likes performing is base leaping and now she is trying to earn money with it. Puerto Rico is exactly where he's been living for many years and he will never transfer. Bookkeeping is my profession.<br><br>my weblog [http://Tomport.ru/node/621726 at home std testing]
	~~\| name = Heawood graph~~
	~~\| image = [[Image:Heawood_Graph.svg\|180px]]~~
	~~\| image_caption =~~
	~~\| namesake = [[Percy John Heawood]]~~
	~~\| vertices = 14~~
	~~\| edges = 21~~
	~~\| girth = 6~~
	~~\| diameter = 3~~
	~~\| radius = 3~~
	~~\| automorphisms = 336 ([[Projective linear group\|PGL<sub>2</sub>(7)]])~~
	~~\| chromatic_number = 2~~
	~~\| chromatic_index = 3~~
	\| properties = [[Bipartite graph\|Bipartite]]<br>[[Cubic graph\|Cubic]]<br>[[Cage (graph theory)\|Cage]]<br>[[Distance-transitive graph\|Distance-transitive]]<br>[[Distance-regular graph\|Distance-regular]]<br>[[Toroidal graph\|Toroidal]]<br>[[Hamiltonian graph\|Hamiltonian]]<br>[[Symmetric graph\|Symmetric]]<br>Orientably simple
	}}

	In the [[mathematics\|mathematical]] field of [[graph theory]], the '''Heawood graph''' is an [[undirected graph]] with 14 vertices and 21 edges, named after [[Percy John Heawood]].<ref>{{MathWorld \| urlname=HeawoodGraph \| title=Heawood Graph}}</ref>

	~~==Combinatorial properties==~~
	The ~~graph~~ is ~~[[cubic graph\|cubic]], and all cycles in~~ the ~~graph have six or more edges. Every smaller cubic graph has shorter cycles, so this graph is the 6-[[cage (graph theory)\|cage]], the smallest cubic graph~~ of ~~[[girth (graph theory)\|girth]] 6~~. It is a [[distance-transitive graph]] (see the [[Foster census]]) and therefore [[Distance-regular graph\|distance regular]].<ref name="brouwer">{{cite web \| author=Brouwer, Andries E. \| authorlink = Andries Brouwer \| title = Heawood graph \| url = http://www.win.tue.nl/~aeb/drg/graphs/Heawood.html \| work = [http://www.win.tue.nl/~aeb/drg/index.html Additions and Corrections to the book ''~~Distance-Regular Graphs''] (Brouwer, Cohen, Neumaier; Springer; 1989)}}</ref>~~

	~~There are 24 [[perfect matching]]~~s ~~in the Heawood graph; for each matching, the set of edges~~ not ~~in the matching forms~~ a [[Hamiltonian cycle]]. For instance, the figure shows the vertices of the graph placed on a cycle, with the internal diagonals of the cycle forming a matching. By subdividing the cycle edges into two matchings, we can partition the Heawood graph into three perfect matchings (that is~~, [[Edge coloring\|3-color its edges]]) in eight different ways.<ref name="brouwer"/> Every two perfect matchings,~~ and ~~every two Hamiltonian cycles, can be transformed into each other by a symmetry of the graph.<ref>{{citation~~
	~~\| last1 = Abreu \| first1 = M.~~
	~~\| last2 = Aldred \| first2 = R. E. L.~~
	~~\| last3 = Funk \| first3 = M.~~
	~~\| last4 = Jackson \| first4 = Bill~~
	~~\| last5 = Labbate \| first5 = D.~~
	~~\| last6 = Sheehan \| first6 = J.~~
	~~\| doi = 10.1016/j.jctb.2004.09.004~~
	~~\| mr = 2099150~~
	~~\| issue = 2~~
	~~\| journal = [[Journal of Combinatorial Theory]] \| series = Series B~~
	~~\| pages = 395–404~~
	~~\| title = Graphs and digraphs~~ with ~~all 2-factors isomorphic~~
	~~\| volume = 92~~
	~~\| year = 2004}}~~.~~</ref>~~

	~~There are 28 six-vertex cycles in the Heawood graph. Each 6-cycle~~ is ~~disjoint from~~ exactly ~~three other 6-cycles; among these three 6-cycles, each one is the symmetric difference of the other two. The graph with one node per 6-cycle, and one edge~~ for each disjoint pair of 6-cycles, is the [[Coxeter graph]].<ref>{{citation\|first=Italo J.\|last=Dejter\|title=From the Coxeter graph to the Klein graph\|journal=Journal of Graph Theory\|year=2011\|doi=10.1002/jgt.20597\|arxiv=1002.1960}}.</ref>

	~~==Geometric and topological properties==~~
	The Heawood graph is a [[toroidal graph]]; that is, it can be embedded without crossings onto a [[torus]]. One embedding of this type places its vertices and edges into three-dimensional [[Euclidean space]] as the set of vertices and edges of a nonconvex polyhedron with the topology of a torus, the [[Szilassi polyhedron]].

	The graph is named after [[Percy John Heawood]], who in 1890 proved that in every subdivision of the torus into polygons, the polygonal regions can be colored by at most seven colors.<ref>{{cite journal \| doi=10.2307/3219140 \| author=Brown, Ezra \| title = The many names of (7,3,1) \| journal = [[Mathematics Magazine]] \| volume=75 \| issue=2 \| year=2002 \| pages=83–94 \| url=http://www.math.vt.edu/people/brown/doc/731.pdf \| jstor=3219140}}</ref><ref>{{cite journal \| author=Heawood, P. J. \| authorlink=Percy John Heawood \| title=Map colouring theorems \| journal = Quarterly J. Math. Oxford Ser. \| year=1890 \| volume=24 \| pages=322–339}}</ref> The Heawood graph forms a subdivision of the torus with seven mutually adjacent regions, showing that this bound is tight.

	The Heawood graph is also the [[Levi graph]] of the [[Fano plane]], the graph representing incidences between points and lines in that geometry. With this interpretation, the 6-cycles in the Heawood graph correspond to [[triangle]]s in the Fano plane.

	~~The Heawood graph has [[Crossing number (graph theory)\|crossing number]] 3,~~ and ~~is the smallest cubic graph with that crossing number {{OEIS\|id=A110507}}. Including the Heawood graph, there are 8 distinct graphs of order 14 with crossing number 3~~.

	~~The Heawood graph~~ is a [[unit distance graph]]: it can be embedded in the plane such that adjacent vertices are exactly at distance one apart, with no two vertices embedded to the same point and no vertex embedded into a point within an edge. However, the known embeddings of this type lack any of the symmetries that are inherent in the graph.<~~ref~~>~~{{Cite journal \| last = Gerbracht \| first = Eberhard H.-A. \| title = Eleven unit distance embeddings of the Heawood graph \| year = 2009 \| arxiv = 0912.5395}}.~~<~~/ref~~>

	~~==Algebraic properties==~~
	~~The [~~[automorphism group]] of the Heawood graph is isomorphic to the [[projective linear group]] PGL<sub>2</sub>(7), a group of order 336.<ref>{{cite book\|last1=Bondy\|first1=J. A.\|author1-link=John Adrian Bondy\|last2=Murty\|first2=U. S. R.\|author2-link=U. S. R. Murty\|title=Graph Theory with Applications\|location=New York\|publisher=North Holland\|year=1976\|page=237\|isbn=0-444-19451-7\|url=http://~~www.ecp6~~.~~jussieu.fr~~/~~pageperso~~/bondy/books/gtwa/gtwa.html}}</ref> It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore the Heawood graph is a [[symmetric graph]]. It has automorphisms that take any vertex to any other vertex and any edge to any other edge. According to the [[Foster census]], the Heawood graph, referenced as F014A, is the only cubic symmetric graph on 14 vertices.<ref>Royle, G. [http://www.cs.uwa.edu.au/~gordon/remote/foster/#census "Cubic Symmetric Graphs (The Foster Census)."]</ref><ref>[[Marston Conder\|Conder, M.]] and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41-63, 2002.</ref>

	~~The [[characteristic polynomial]] of the Heawood graph is <math>(x-3) (x+3) (x^2-2)^6</math>. It is the only graph with this characteristic polynomial, making it a graph determined by its spectrum.~~

	~~==Gallery==~~
	~~<gallery>~~
	~~Image:Szilassi polyhedron.svg\|The [[Szilassi polyhedron]].~~
	~~File:3-crossing Heawood graph.svg\|The Heawood graph has [[Crossing number (graph theory)\|crossing number]] 3.~~
	~~File:Heawood graph 3color edge.svg\|The [[chromatic index]] of the Heawood graph is 3.~~
	~~File:Heawood graph 2COL.svg\|The [[chromatic number]] of the Heawood graph is 2.~~
	~~File:7x-torus.svg\|An embedding of the Heawood graph onto a torus (shown as a square with [[periodic boundary conditions]]) partitioning it into seven mutually-adjacent regions~~
	~~</gallery>~~

	~~== References ==~~
	~~{{reflist}}~~

	~~[[Category:Individual graphs]]~~
	~~[[Category:Regular graphs]~~]

220.239.119.190 at 09:37, 21 September 2013

2013-09-21T09:37:45Z

← Older revision		Revision as of 11:37, 21 September 2013
Line 1:		Line 1:
			{{infobox graph
			\| name = Heawood graph
			\| image = [[Image:Heawood_Graph.svg\|180px]]
			\| image_caption =
			\| namesake = [[Percy John Heawood]]
			\| vertices = 14
			\| edges = 21
			\| girth = 6
			\| diameter = 3
			\| radius = 3
			\| automorphisms = 336 ([[Projective linear group\|PGL<sub>2</sub>(7)]])
			\| chromatic_number = 2
			\| chromatic_index = 3
			\| properties = [[Bipartite graph\|Bipartite]]<br>[[Cubic graph\|Cubic]]<br>[[Cage (graph theory)\|Cage]]<br>[[Distance-transitive graph\|Distance-transitive]]<br>[[Distance-regular graph\|Distance-regular]]<br>[[Toroidal graph\|Toroidal]]<br>[[Hamiltonian graph\|Hamiltonian]]<br>[[Symmetric graph\|Symmetric]]<br>Orientably simple
			}}

			In the [[mathematics\|mathematical]] field of [[graph theory]], the '''Heawood graph''' is an [[undirected graph]] with 14 vertices and 21 edges, named after [[Percy John Heawood]].<ref>{{MathWorld \| urlname=HeawoodGraph \| title=Heawood Graph}}</ref>

	~~Hello person~~. ~~ʟet me introduce myself~~. ~~I am Coral Gehl but my spouse doesn~~'~~t ѕimilar to it at virtually аll~~. ~~Minnesota іs where heг property~~ is ~~bսt she needs~~ to ~~flex bеcause~~ of ~~her cherished ones~~. One of ~~tɦe~~ most [http://www.~~Dailymail~~.co.uk/~~home~~/~~search~~.~~html?sel~~=~~site&searchPhrase~~=~~notable notable~~] ~~thіngs~~ in the ~~arena for Һеr~~ is ~~cycling and sҺe'll ƅе starting somethіng еlse аlong utilizing~~. ~~Interviewing~~ is ~~where hіs primary income ɡet fгom~~. ~~Sɦe's not gоod at design anyone might~~ in ~~oгder~~ to ~~check her website:~~ http://~~antoi27b~~.~~buzznet~~.~~com~~/~~user~~/~~journal~~/~~18347949~~/~~teach-casino-p-norsk-better~~/<br><br>~~Also visit my homepage ::~~ [http://~~antoi27b~~.~~buzznet~~.~~com~~/~~user~~/~~journal~~/~~18347949~~/~~teach~~-~~casino~~-p-~~norsk~~-~~better~~/ ~~the very top online Casino room~~]		==Combinatorial properties==
			The graph is [[cubic graph\|cubic]], and all cycles in the graph have six or more edges. Every smaller cubic graph has shorter cycles, so this graph is the 6-[[cage (graph theory)\|cage]], the smallest cubic graph of [[girth (graph theory)\|girth]] 6. It is a [[distance-transitive graph]] (see the [[Foster census]]) and therefore [[Distance-regular graph\|distance regular]].<ref name="brouwer">{{cite web \| author=Brouwer, Andries E. \| authorlink = Andries Brouwer \| title = Heawood graph \| url = http://www.win.tue.nl/~aeb/drg/graphs/Heawood.html \| work = [http://www.win.tue.nl/~aeb/drg/index.html Additions and Corrections to the book ''Distance-Regular Graphs''] (Brouwer, Cohen, Neumaier; Springer; 1989)}}</ref>

			There are 24 [[perfect matching]]s in the Heawood graph; for each matching, the set of edges not in the matching forms a [[Hamiltonian cycle]]. For instance, the figure shows the vertices of the graph placed on a cycle, with the internal diagonals of the cycle forming a matching. By subdividing the cycle edges into two matchings, we can partition the Heawood graph into three perfect matchings (that is, [[Edge coloring\|3-color its edges]]) in eight different ways.<ref name="brouwer"/> Every two perfect matchings, and every two Hamiltonian cycles, can be transformed into each other by a symmetry of the graph.<ref>{{citation
			\| last1 = Abreu \| first1 = M.
			\| last2 = Aldred \| first2 = R. E. L.
			\| last3 = Funk \| first3 = M.
			\| last4 = Jackson \| first4 = Bill
			\| last5 = Labbate \| first5 = D.
			\| last6 = Sheehan \| first6 = J.
			\| doi = 10.1016/j.jctb.2004.09.004
			\| mr = 2099150
			\| issue = 2
			\| journal = [[Journal of Combinatorial Theory]] \| series = Series B
			\| pages = 395–404
			\| title = Graphs and digraphs with all 2-factors isomorphic
			\| volume = 92
			\| year = 2004}}.</ref>

			There are 28 six-vertex cycles in the Heawood graph. Each 6-cycle is disjoint from exactly three other 6-cycles; among these three 6-cycles, each one is the symmetric difference of the other two. The graph with one node per 6-cycle, and one edge for each disjoint pair of 6-cycles, is the [[Coxeter graph]].<ref>{{citation\|first=Italo J.\|last=Dejter\|title=From the Coxeter graph to the Klein graph\|journal=Journal of Graph Theory\|year=2011\|doi=10.1002/jgt.20597\|arxiv=1002.1960}}.</ref>

			==Geometric and topological properties==
			The Heawood graph is a [[toroidal graph]]; that is, it can be embedded without crossings onto a [[torus]]. One embedding of this type places its vertices and edges into three-dimensional [[Euclidean space]] as the set of vertices and edges of a nonconvex polyhedron with the topology of a torus, the [[Szilassi polyhedron]].

			The graph is named after [[Percy John Heawood]], who in 1890 proved that in every subdivision of the torus into polygons, the polygonal regions can be colored by at most seven colors.<ref>{{cite journal \| doi=10.2307/3219140 \| author=Brown, Ezra \| title = The many names of (7,3,1) \| journal = [[Mathematics Magazine]] \| volume=75 \| issue=2 \| year=2002 \| pages=83–94 \| url=http://www.math.vt.edu/people/brown/doc/731.pdf \| jstor=3219140}}</ref><ref>{{cite journal \| author=Heawood, P. J. \| authorlink=Percy John Heawood \| title=Map colouring theorems \| journal = Quarterly J. Math. Oxford Ser. \| year=1890 \| volume=24 \| pages=322–339}}</ref> The Heawood graph forms a subdivision of the torus with seven mutually adjacent regions, showing that this bound is tight.

			The Heawood graph is also the [[Levi graph]] of the [[Fano plane]], the graph representing incidences between points and lines in that geometry. With this interpretation, the 6-cycles in the Heawood graph correspond to [[triangle]]s in the Fano plane.

			The Heawood graph has [[Crossing number (graph theory)\|crossing number]] 3, and is the smallest cubic graph with that crossing number {{OEIS\|id=A110507}}. Including the Heawood graph, there are 8 distinct graphs of order 14 with crossing number 3.

			The Heawood graph is a [[unit distance graph]]: it can be embedded in the plane such that adjacent vertices are exactly at distance one apart, with no two vertices embedded to the same point and no vertex embedded into a point within an edge. However, the known embeddings of this type lack any of the symmetries that are inherent in the graph.<ref>{{Cite journal \| last = Gerbracht \| first = Eberhard H.-A. \| title = Eleven unit distance embeddings of the Heawood graph \| year = 2009 \| arxiv = 0912.5395}}.</ref>

			==Algebraic properties==
			The [[automorphism group]] of the Heawood graph is isomorphic to the [[projective linear group]] PGL<sub>2</sub>(7), a group of order 336.<ref>{{cite book\|last1=Bondy\|first1=J. A.\|author1-link=John Adrian Bondy\|last2=Murty\|first2=U. S. R.\|author2-link=U. S. R. Murty\|title=Graph Theory with Applications\|location=New York\|publisher=North Holland\|year=1976\|page=237\|isbn=0-444-19451-7\|url=http://www.ecp6.jussieu.fr/pageperso/bondy/books/gtwa/gtwa.html}}</ref> It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore the Heawood graph is a [[symmetric graph]]. It has automorphisms that take any vertex to any other vertex and any edge to any other edge. According to the [[Foster census]], the Heawood graph, referenced as F014A, is the only cubic symmetric graph on 14 vertices.<ref>Royle, G. [http://www.cs.uwa.edu.au/~gordon/remote/foster/#census "Cubic Symmetric Graphs (The Foster Census)."]</ref><ref>[[Marston Conder\|Conder, M.]] and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41-63, 2002.</ref>

			The [[characteristic polynomial]] of the Heawood graph is <math>(x-3) (x+3) (x^2-2)^6</math>. It is the only graph with this characteristic polynomial, making it a graph determined by its spectrum.

			==Gallery==
			<gallery>
			Image:Szilassi polyhedron.svg\|The [[Szilassi polyhedron]].
			File:3-crossing Heawood graph.svg\|The Heawood graph has [[Crossing number (graph theory)\|crossing number]] 3.
			File:Heawood graph 3color edge.svg\|The [[chromatic index]] of the Heawood graph is 3.
			File:Heawood graph 2COL.svg\|The [[chromatic number]] of the Heawood graph is 2.
			File:7x-torus.svg\|An embedding of the Heawood graph onto a torus (shown as a square with [[periodic boundary conditions]]) partitioning it into seven mutually-adjacent regions
			</gallery>

			== References ==
			{{reflist}}

			[[Category:Individual graphs]]
			[[Category:Regular graphs]]

en>EmausBot: r2.7.2+) (Robot: Adding be:Модуль адлегласці

2012-08-23T18:27:21Z

r2.7.2+) (Robot: Adding be:Модуль адлегласці

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← Older revision		Revision as of 23:36, 7 February 2014
Line 1:		Line 1:
	~~{{infobox graph~~		The author is recognized by the name of Numbers Lint. It's not a typical thing but what she likes performing is base leaping and now she is trying to earn money with it. Puerto Rico is exactly where he's been living for many years and he will never transfer. Bookkeeping is my profession.<br><br>my weblog [http://Tomport.ru/node/621726 at home std testing]
	~~\| name = Heawood graph~~
	~~\| image = [[Image:Heawood_Graph.svg\|180px]]~~
	~~\| image_caption =~~
	~~\| namesake = [[Percy John Heawood]]~~
	~~\| vertices = 14~~
	~~\| edges = 21~~
	~~\| girth = 6~~
	~~\| diameter = 3~~
	~~\| radius = 3~~
	~~\| automorphisms = 336 ([[Projective linear group\|PGL<sub>2</sub>(7)]])~~
	~~\| chromatic_number = 2~~
	~~\| chromatic_index = 3~~
	\| properties = [[Bipartite graph\|Bipartite]]<br>[[Cubic graph\|Cubic]]<br>[[Cage (graph theory)\|Cage]]<br>[[Distance-transitive graph\|Distance-transitive]]<br>[[Distance-regular graph\|Distance-regular]]<br>[[Toroidal graph\|Toroidal]]<br>[[Hamiltonian graph\|Hamiltonian]]<br>[[Symmetric graph\|Symmetric]]<br>Orientably simple
	}}

	In the [[mathematics\|mathematical]] field of [[graph theory]], the '''Heawood graph''' is an [[undirected graph]] with 14 vertices and 21 edges, named after [[Percy John Heawood]].<ref>{{MathWorld \| urlname=HeawoodGraph \| title=Heawood Graph}}</ref>

	~~==Combinatorial properties==~~
	The ~~graph~~ is ~~[[cubic graph\|cubic]], and all cycles in~~ the ~~graph have six or more edges. Every smaller cubic graph has shorter cycles, so this graph is the 6-[[cage (graph theory)\|cage]], the smallest cubic graph~~ of ~~[[girth (graph theory)\|girth]] 6~~. It is a [[distance-transitive graph]] (see the [[Foster census]]) and therefore [[Distance-regular graph\|distance regular]].<ref name="brouwer">{{cite web \| author=Brouwer, Andries E. \| authorlink = Andries Brouwer \| title = Heawood graph \| url = http://www.win.tue.nl/~aeb/drg/graphs/Heawood.html \| work = [http://www.win.tue.nl/~aeb/drg/index.html Additions and Corrections to the book ''~~Distance-Regular Graphs''] (Brouwer, Cohen, Neumaier; Springer; 1989)}}</ref>~~

	~~There are 24 [[perfect matching]]~~s ~~in the Heawood graph; for each matching, the set of edges~~ not ~~in the matching forms~~ a [[Hamiltonian cycle]]. For instance, the figure shows the vertices of the graph placed on a cycle, with the internal diagonals of the cycle forming a matching. By subdividing the cycle edges into two matchings, we can partition the Heawood graph into three perfect matchings (that is~~, [[Edge coloring\|3-color its edges]]) in eight different ways.<ref name="brouwer"/> Every two perfect matchings,~~ and ~~every two Hamiltonian cycles, can be transformed into each other by a symmetry of the graph.<ref>{{citation~~
	~~\| last1 = Abreu \| first1 = M.~~
	~~\| last2 = Aldred \| first2 = R. E. L.~~
	~~\| last3 = Funk \| first3 = M.~~
	~~\| last4 = Jackson \| first4 = Bill~~
	~~\| last5 = Labbate \| first5 = D.~~
	~~\| last6 = Sheehan \| first6 = J.~~
	~~\| doi = 10.1016/j.jctb.2004.09.004~~
	~~\| mr = 2099150~~
	~~\| issue = 2~~
	~~\| journal = [[Journal of Combinatorial Theory]] \| series = Series B~~
	~~\| pages = 395–404~~
	~~\| title = Graphs and digraphs~~ with ~~all 2-factors isomorphic~~
	~~\| volume = 92~~
	~~\| year = 2004}}~~.~~</ref>~~

	~~There are 28 six-vertex cycles in the Heawood graph. Each 6-cycle~~ is ~~disjoint from~~ exactly ~~three other 6-cycles; among these three 6-cycles, each one is the symmetric difference of the other two. The graph with one node per 6-cycle, and one edge~~ for each disjoint pair of 6-cycles, is the [[Coxeter graph]].<ref>{{citation\|first=Italo J.\|last=Dejter\|title=From the Coxeter graph to the Klein graph\|journal=Journal of Graph Theory\|year=2011\|doi=10.1002/jgt.20597\|arxiv=1002.1960}}.</ref>

	~~==Geometric and topological properties==~~
	The Heawood graph is a [[toroidal graph]]; that is, it can be embedded without crossings onto a [[torus]]. One embedding of this type places its vertices and edges into three-dimensional [[Euclidean space]] as the set of vertices and edges of a nonconvex polyhedron with the topology of a torus, the [[Szilassi polyhedron]].

	The graph is named after [[Percy John Heawood]], who in 1890 proved that in every subdivision of the torus into polygons, the polygonal regions can be colored by at most seven colors.<ref>{{cite journal \| doi=10.2307/3219140 \| author=Brown, Ezra \| title = The many names of (7,3,1) \| journal = [[Mathematics Magazine]] \| volume=75 \| issue=2 \| year=2002 \| pages=83–94 \| url=http://www.math.vt.edu/people/brown/doc/731.pdf \| jstor=3219140}}</ref><ref>{{cite journal \| author=Heawood, P. J. \| authorlink=Percy John Heawood \| title=Map colouring theorems \| journal = Quarterly J. Math. Oxford Ser. \| year=1890 \| volume=24 \| pages=322–339}}</ref> The Heawood graph forms a subdivision of the torus with seven mutually adjacent regions, showing that this bound is tight.

	The Heawood graph is also the [[Levi graph]] of the [[Fano plane]], the graph representing incidences between points and lines in that geometry. With this interpretation, the 6-cycles in the Heawood graph correspond to [[triangle]]s in the Fano plane.

	~~The Heawood graph has [[Crossing number (graph theory)\|crossing number]] 3,~~ and ~~is the smallest cubic graph with that crossing number {{OEIS\|id=A110507}}. Including the Heawood graph, there are 8 distinct graphs of order 14 with crossing number 3~~.

	~~The Heawood graph~~ is a [[unit distance graph]]: it can be embedded in the plane such that adjacent vertices are exactly at distance one apart, with no two vertices embedded to the same point and no vertex embedded into a point within an edge. However, the known embeddings of this type lack any of the symmetries that are inherent in the graph.<~~ref~~>~~{{Cite journal \| last = Gerbracht \| first = Eberhard H.-A. \| title = Eleven unit distance embeddings of the Heawood graph \| year = 2009 \| arxiv = 0912.5395}}.~~<~~/ref~~>

	~~==Algebraic properties==~~
	~~The [~~[automorphism group]] of the Heawood graph is isomorphic to the [[projective linear group]] PGL<sub>2</sub>(7), a group of order 336.<ref>{{cite book\|last1=Bondy\|first1=J. A.\|author1-link=John Adrian Bondy\|last2=Murty\|first2=U. S. R.\|author2-link=U. S. R. Murty\|title=Graph Theory with Applications\|location=New York\|publisher=North Holland\|year=1976\|page=237\|isbn=0-444-19451-7\|url=http://~~www.ecp6~~.~~jussieu.fr~~/~~pageperso~~/bondy/books/gtwa/gtwa.html}}</ref> It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore the Heawood graph is a [[symmetric graph]]. It has automorphisms that take any vertex to any other vertex and any edge to any other edge. According to the [[Foster census]], the Heawood graph, referenced as F014A, is the only cubic symmetric graph on 14 vertices.<ref>Royle, G. [http://www.cs.uwa.edu.au/~gordon/remote/foster/#census "Cubic Symmetric Graphs (The Foster Census)."]</ref><ref>[[Marston Conder\|Conder, M.]] and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41-63, 2002.</ref>

	~~The [[characteristic polynomial]] of the Heawood graph is <math>(x-3) (x+3) (x^2-2)^6</math>. It is the only graph with this characteristic polynomial, making it a graph determined by its spectrum.~~

	~~==Gallery==~~
	~~<gallery>~~
	~~Image:Szilassi polyhedron.svg\|The [[Szilassi polyhedron]].~~
	~~File:3-crossing Heawood graph.svg\|The Heawood graph has [[Crossing number (graph theory)\|crossing number]] 3.~~
	~~File:Heawood graph 3color edge.svg\|The [[chromatic index]] of the Heawood graph is 3.~~
	~~File:Heawood graph 2COL.svg\|The [[chromatic number]] of the Heawood graph is 2.~~
	~~File:7x-torus.svg\|An embedding of the Heawood graph onto a torus (shown as a square with [[periodic boundary conditions]]) partitioning it into seven mutually-adjacent regions~~
	~~</gallery>~~

	~~== References ==~~
	~~{{reflist}}~~

	~~[[Category:Individual graphs]]~~
	~~[[Category:Regular graphs]~~]