31.221.13.140 at 16:28, 17 November 2014

2014-11-17T16:28:45Z

131.251.253.7: /* Semipartial correlation (part correlation) */

2014-02-13T08:23:02Z

Semipartial correlation (part correlation)

← Older revision		Revision as of 10:23, 13 February 2014
Line 1:		Line 1:
	~~In [[graph theory\|graph-theoretic mathematics]]~~, ~~a '''voltage graph''' is a [[directed graph]] whose edges are labelled invertibly by elements of a [[group (mathematics)\|group]]. It is formally identical to a [[gain graph]], but it~~ is ~~generally used in [[topological graph theory]] as a concise way to specify another [[graph (mathematics)\|graph]] called the [[derived graph]] of the voltage graph.~~		Hello, my title is Andrew and my spouse doesn't like it at all. One of the things she enjoys most is canoeing and she's been performing it for fairly a while. Invoicing is what I do. Mississippi is the only place I've been residing in but I will have to transfer in a yr or two.<br><br>Also visit my web page: free tarot readings ([http://chorokdeul.co.kr/index.php?document_srl=324263&mid=customer21 chorokdeul.co.kr])

	Typical choices of the groups used for voltage graphs include the two-element group ℤ<sub>2</sub> (for defining the [[bipartite double cover]] of a graph), [[free group]]s (for defining the [[covering graph\|universal cover]] of a graph), ''d''-dimensional [[integer lattice]]s ℤ<sup>''d''</sup> (viewed as a group under vector addition, for defining periodic structures in ''d''-dimensional [[Euclidean space]]),<ref name="periodic">{{harvtxt\|Iwano\|Steiglitz\|1987}}; {{harvtxt\|Kosaraju\|Sullivan\|1988}}; {{harvtxt\|Cohen\|Megiddo\|1989}}.</ref> and ~~finite [[cyclic group]]s ℤ<sub>~~'~~'n''</sub> for ''n'' > 2~~. ~~When Π is a cyclic group, the voltage graph may be called a ''cyclic-voltage graph''.~~

	~~==Definition==~~
	~~Formal definition of a <span class="texhtml">Π</span>-voltage graph, for a given group <span class="texhtml">Π</span>:~~
	* Begin with a [[digraph (mathematics)\|digraph]] ''G''. (The direction is solely for convenience in notation.)
	* A <span class="texhtml">Π</span>-voltage on an arc of ''G'' is a label of the ~~arc by an element ''x'' of <span class="texhtml">Π</span>. For instance, in the case where <span class="texhtml">Π = ℤ<sub>''n''</sub></span>, the label~~ is ~~a number ''i'' (mod ''n'').~~
	* A <span class="texhtml">Π</span>-voltage assignment is a function <math>\alpha : E(G) \rightarrow \Pi</math> that labels each arc of ''G'' with a Π-voltage.
	* A <span class="texhtml">Π</span>-voltage graph is a pair <math>( G, \alpha: E(G) \rightarrow \Pi )</math> such that ''G'' is a digraph and ~~α is a voltage assignment.~~
	* The '~~''voltage group''' of~~ a ~~voltage graph <math>( G, \alpha: E(G) \rightarrow \Pi )</math> is the group <span class="texhtml">Π</span> from which the voltages are assigned~~.

	~~Note that the voltages of a voltage graph need not satisfy [[Kirchhoff's circuit laws#Kirchhoff's voltage law (KVL)\|Kirchhoff's voltage law]], that the sum of voltages around a closed path~~ is ~~0 (the identity element of the group), although this law does hold for the derived graphs described below. Thus, the name may be somewhat misleading~~. ~~It results from the origin of voltage graphs as dual to the [[current graph]]s of [[topological graph theory]].~~

	~~==The derived graph==~~
	~~The '''derived graph''' of a voltage graph <math>( G, \alpha: E(G) \rightarrow \mathbb{Z}_{n} )</math>~~ is the graph <math>\tilde G</math> whose vertex set is <math>\tilde V = V \times \mathbb{Z}_{n}</math> and whose edge set is <math>\tilde E = E \times \mathbb{Z}_{n}</math>, where the endpoints of an edge (''e'', ''k'') such that ''e'' has tail ''v'' and head '~~'w'' are <math>(v,\ k)</math> and <math>(w,\ k+\alpha(e))</math>.~~

	Although voltage graphs are defined for digraphs, they may be extended to [[undirected graph]]s by replacing each undirected edge by a pair of oppositely ordered directed edges and by requiring that these edges have labels that are inverse to each other in ~~the group structure. In this case, the derived graph~~ will ~~also~~ have ~~the property that its directed edges form pairs of oppositely oriented edges, so the derived graph may itself be interpreted as being an undirected graph.~~

	The derived graph is a [[covering graph]] of the given voltage graph. If no edge label of the voltage graph is the identity element, then the group elements associated with the vertices of the derived graph provide a [[graph coloring\|coloring]] of the derived graph with a number of colors equal to ~~the group order. An important special case is the [[bipartite double cover]], the derived graph of a voltage graph~~ in ~~which all edges are labeled with the non-identity element of~~ a two~~-element group. Because the order of the group is two, the derived graph in this case is guaranteed to be [[bipartite graph\|bipartite]]~~.

	~~[[Polynomial time]] [[algorithm]]s are known for determining whether the derived graph of a~~ <~~math~~>~~\mathbb{Z}^d~~<~~/math~~>~~-voltage graph contains any directed cycles.<ref name="periodic"/>~~

	~~==Examples==~~
	Any [[Cayley graph]] of a group Π, with a given set Γ of generators, may be defined as the derived graph for a Π-voltage graph having one vertex and \|Γ\| self-loops, each labeled with one of the generators in Γ.<ref>{{harvtxt\|Gross\|Tucker\|1987}}, Theorem 2.2.3, p. 69.</ref>

	~~The [[Petersen graph]] is the derived graph for a ℤ<sub>5</sub>-voltage graph in the shape of a dumbbell with two vertices and three edges~~: one edge connecting the two vertices, and one self-loop on each vertex. One self-loop is labeled with 1, the other with 2, and the edge connecting the two vertices is labeled 0. More generally, the same construction allows any [[generalized Petersen graph]] GP(~~''n'',''k'') to be constructed as a derived graph of the same dumbbell graph with labels 1, 0, and ''k'' in the group ℤ<sub>''n''</sub>.<ref>{{harvtxt\|Gross\|Tucker\|1987}}, Example 2.1.2, p.58.</ref>~~

	~~The vertices and edges of any periodic~~ [~~[tesellation]] of the plane may be formed as the derived graph of a finite graph, with voltages in ℤ<sup>2<~~/~~sup>.~~

	~~==Notes==~~
	~~{{reflist}}~~

	~~==References==~~
	*{{citation
	~~\| last1 = Cohen \| first1 = Edith~~
	~~\| last2 = Megiddo \| first2 = Nimrod~~
	~~\|authorlink2= Nimrod Megiddo~~
	~~\| contribution = Strongly polynomial-time and NC algorithms for detecting cycles in dynamic graphs~~
	~~\| doi = 10.1145~~/~~73007~~.~~73057~~
	~~\| pages = 523–534~~
	~~\| title = Proc. 21st Annual ACM Symposium on Theory of Computing (STOC '89)~~
	~~\| year = 1989}}~~.
	*{{citation
	~~\| last = Gross \| first = Jonathan L.~~
	~~\| doi = 10.1016~~/~~0012-365X(74)90006-5~~
	~~\| issue = 3~~
	~~\| journal = Discrete Mathematics~~
	~~\| pages = 239–246~~
	~~\| title = Voltage graphs~~
	~~\| volume = 9~~
	~~\| year = 1974}}~~.
	*{{citation
	~~\| last1~~ = ~~Gross \| first1~~ = ~~Jonathan L~~.
	~~\| last2 = Tucker \| first2 = Thomas W~~.
	~~\| doi = 10.1016/0012-365X(77)90131-5~~
	~~\| journal = Discrete Mathematics~~
	~~\| pages = 273–283~~
	~~\| title = Generating all graph coverings by permutation voltage assignments~~
	~~\| issue = 3~~
	~~\| volume = 18~~
	~~\| year = 1977}}.~~
	*{{citation
	~~\| last1 = Gross \| first1 = Jonathan L.~~
	~~\| last2 = Tucker \| first2 = Thomas W.~~
	~~\| location = New York~~
	~~\| publisher = Wiley~~
	~~\| title = Topological Graph Theory~~
	~~\| year = 1987}}.~~
	*{{citation
	~~\| last1 = Iwano \| first1 = K.~~
	~~\| last2 = Steiglitz \| first2 = K.~~
	~~\| contribution = Testing for cycles in infinite graphs with periodic structure~~
	~~\| doi = 10.1145/28395.28401~~
	~~\| pages = 46–55~~
	~~\| title = Proc. 19th Annual ACM Symposium on Theory of Computing (STOC '87~~)
	~~\| year = 1987}}.~~
	*{{citation
	~~\| last1 = Kosaraju \| first1 = S. Rao \| author1-link = S. Rao Kosaraju~~
	~~\| last2 = Sullivan \| first2 = Gregory~~
	~~\| contribution = Detecting cycles in dynamic graphs in polynomial time~~
	~~\| doi = 10.1145/62212.62251~~
	~~\| pages = 398–406~~
	~~\| title = Proc. 20th Annual ACM Symposium on Theory of Computing (STOC '88)~~
	~~\| year = 1988}}.~~

	~~[[Category:Extensions and generalizations of graphs]]~~

en>Terhorst: /* Using matrix inversion */

2013-12-16T16:20:57Z

Using matrix inversion

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en>GoingBatty: WPCleaner v1.13 - Repaired 1 link to disambiguation page - (You can help) - Metron

2012-06-27T01:42:44Z

WPCleaner v1.13 - Repaired 1 link to disambiguation page - (You can help) - Metron

Show changes

← Older revision		Revision as of 10:23, 13 February 2014
Line 1:		Line 1:
	~~In [[graph theory\|graph-theoretic mathematics]]~~, ~~a '''voltage graph''' is a [[directed graph]] whose edges are labelled invertibly by elements of a [[group (mathematics)\|group]]. It is formally identical to a [[gain graph]], but it~~ is ~~generally used in [[topological graph theory]] as a concise way to specify another [[graph (mathematics)\|graph]] called the [[derived graph]] of the voltage graph.~~		Hello, my title is Andrew and my spouse doesn't like it at all. One of the things she enjoys most is canoeing and she's been performing it for fairly a while. Invoicing is what I do. Mississippi is the only place I've been residing in but I will have to transfer in a yr or two.<br><br>Also visit my web page: free tarot readings ([http://chorokdeul.co.kr/index.php?document_srl=324263&mid=customer21 chorokdeul.co.kr])

	Typical choices of the groups used for voltage graphs include the two-element group ℤ<sub>2</sub> (for defining the [[bipartite double cover]] of a graph), [[free group]]s (for defining the [[covering graph\|universal cover]] of a graph), ''d''-dimensional [[integer lattice]]s ℤ<sup>''d''</sup> (viewed as a group under vector addition, for defining periodic structures in ''d''-dimensional [[Euclidean space]]),<ref name="periodic">{{harvtxt\|Iwano\|Steiglitz\|1987}}; {{harvtxt\|Kosaraju\|Sullivan\|1988}}; {{harvtxt\|Cohen\|Megiddo\|1989}}.</ref> and ~~finite [[cyclic group]]s ℤ<sub>~~'~~'n''</sub> for ''n'' > 2~~. ~~When Π is a cyclic group, the voltage graph may be called a ''cyclic-voltage graph''.~~

	~~==Definition==~~
	~~Formal definition of a <span class="texhtml">Π</span>-voltage graph, for a given group <span class="texhtml">Π</span>:~~
	* Begin with a [[digraph (mathematics)\|digraph]] ''G''. (The direction is solely for convenience in notation.)
	* A <span class="texhtml">Π</span>-voltage on an arc of ''G'' is a label of the ~~arc by an element ''x'' of <span class="texhtml">Π</span>. For instance, in the case where <span class="texhtml">Π = ℤ<sub>''n''</sub></span>, the label~~ is ~~a number ''i'' (mod ''n'').~~
	* A <span class="texhtml">Π</span>-voltage assignment is a function <math>\alpha : E(G) \rightarrow \Pi</math> that labels each arc of ''G'' with a Π-voltage.
	* A <span class="texhtml">Π</span>-voltage graph is a pair <math>( G, \alpha: E(G) \rightarrow \Pi )</math> such that ''G'' is a digraph and ~~α is a voltage assignment.~~
	* The '~~''voltage group''' of~~ a ~~voltage graph <math>( G, \alpha: E(G) \rightarrow \Pi )</math> is the group <span class="texhtml">Π</span> from which the voltages are assigned~~.

	~~Note that the voltages of a voltage graph need not satisfy [[Kirchhoff's circuit laws#Kirchhoff's voltage law (KVL)\|Kirchhoff's voltage law]], that the sum of voltages around a closed path~~ is ~~0 (the identity element of the group), although this law does hold for the derived graphs described below. Thus, the name may be somewhat misleading~~. ~~It results from the origin of voltage graphs as dual to the [[current graph]]s of [[topological graph theory]].~~

	~~==The derived graph==~~
	~~The '''derived graph''' of a voltage graph <math>( G, \alpha: E(G) \rightarrow \mathbb{Z}_{n} )</math>~~ is the graph <math>\tilde G</math> whose vertex set is <math>\tilde V = V \times \mathbb{Z}_{n}</math> and whose edge set is <math>\tilde E = E \times \mathbb{Z}_{n}</math>, where the endpoints of an edge (''e'', ''k'') such that ''e'' has tail ''v'' and head '~~'w'' are <math>(v,\ k)</math> and <math>(w,\ k+\alpha(e))</math>.~~

	Although voltage graphs are defined for digraphs, they may be extended to [[undirected graph]]s by replacing each undirected edge by a pair of oppositely ordered directed edges and by requiring that these edges have labels that are inverse to each other in ~~the group structure. In this case, the derived graph~~ will ~~also~~ have ~~the property that its directed edges form pairs of oppositely oriented edges, so the derived graph may itself be interpreted as being an undirected graph.~~

	The derived graph is a [[covering graph]] of the given voltage graph. If no edge label of the voltage graph is the identity element, then the group elements associated with the vertices of the derived graph provide a [[graph coloring\|coloring]] of the derived graph with a number of colors equal to ~~the group order. An important special case is the [[bipartite double cover]], the derived graph of a voltage graph~~ in ~~which all edges are labeled with the non-identity element of~~ a two~~-element group. Because the order of the group is two, the derived graph in this case is guaranteed to be [[bipartite graph\|bipartite]]~~.

	~~[[Polynomial time]] [[algorithm]]s are known for determining whether the derived graph of a~~ <~~math~~>~~\mathbb{Z}^d~~<~~/math~~>~~-voltage graph contains any directed cycles.<ref name="periodic"/>~~

	~~==Examples==~~
	Any [[Cayley graph]] of a group Π, with a given set Γ of generators, may be defined as the derived graph for a Π-voltage graph having one vertex and \|Γ\| self-loops, each labeled with one of the generators in Γ.<ref>{{harvtxt\|Gross\|Tucker\|1987}}, Theorem 2.2.3, p. 69.</ref>

	~~The [[Petersen graph]] is the derived graph for a ℤ<sub>5</sub>-voltage graph in the shape of a dumbbell with two vertices and three edges~~: one edge connecting the two vertices, and one self-loop on each vertex. One self-loop is labeled with 1, the other with 2, and the edge connecting the two vertices is labeled 0. More generally, the same construction allows any [[generalized Petersen graph]] GP(~~''n'',''k'') to be constructed as a derived graph of the same dumbbell graph with labels 1, 0, and ''k'' in the group ℤ<sub>''n''</sub>.<ref>{{harvtxt\|Gross\|Tucker\|1987}}, Example 2.1.2, p.58.</ref>~~

	~~The vertices and edges of any periodic~~ [~~[tesellation]] of the plane may be formed as the derived graph of a finite graph, with voltages in ℤ<sup>2<~~/~~sup>.~~

	~~==Notes==~~
	~~{{reflist}}~~

	~~==References==~~
	*{{citation
	~~\| last1 = Cohen \| first1 = Edith~~
	~~\| last2 = Megiddo \| first2 = Nimrod~~
	~~\|authorlink2= Nimrod Megiddo~~
	~~\| contribution = Strongly polynomial-time and NC algorithms for detecting cycles in dynamic graphs~~
	~~\| doi = 10.1145~~/~~73007~~.~~73057~~
	~~\| pages = 523–534~~
	~~\| title = Proc. 21st Annual ACM Symposium on Theory of Computing (STOC '89)~~
	~~\| year = 1989}}~~.
	*{{citation
	~~\| last = Gross \| first = Jonathan L.~~
	~~\| doi = 10.1016~~/~~0012-365X(74)90006-5~~
	~~\| issue = 3~~
	~~\| journal = Discrete Mathematics~~
	~~\| pages = 239–246~~
	~~\| title = Voltage graphs~~
	~~\| volume = 9~~
	~~\| year = 1974}}~~.
	*{{citation
	~~\| last1~~ = ~~Gross \| first1~~ = ~~Jonathan L~~.
	~~\| last2 = Tucker \| first2 = Thomas W~~.
	~~\| doi = 10.1016/0012-365X(77)90131-5~~
	~~\| journal = Discrete Mathematics~~
	~~\| pages = 273–283~~
	~~\| title = Generating all graph coverings by permutation voltage assignments~~
	~~\| issue = 3~~
	~~\| volume = 18~~
	~~\| year = 1977}}.~~
	*{{citation
	~~\| last1 = Gross \| first1 = Jonathan L.~~
	~~\| last2 = Tucker \| first2 = Thomas W.~~
	~~\| location = New York~~
	~~\| publisher = Wiley~~
	~~\| title = Topological Graph Theory~~
	~~\| year = 1987}}.~~
	*{{citation
	~~\| last1 = Iwano \| first1 = K.~~
	~~\| last2 = Steiglitz \| first2 = K.~~
	~~\| contribution = Testing for cycles in infinite graphs with periodic structure~~
	~~\| doi = 10.1145/28395.28401~~
	~~\| pages = 46–55~~
	~~\| title = Proc. 19th Annual ACM Symposium on Theory of Computing (STOC '87~~)
	~~\| year = 1987}}.~~
	*{{citation
	~~\| last1 = Kosaraju \| first1 = S. Rao \| author1-link = S. Rao Kosaraju~~
	~~\| last2 = Sullivan \| first2 = Gregory~~
	~~\| contribution = Detecting cycles in dynamic graphs in polynomial time~~
	~~\| doi = 10.1145/62212.62251~~
	~~\| pages = 398–406~~
	~~\| title = Proc. 20th Annual ACM Symposium on Theory of Computing (STOC '88)~~
	~~\| year = 1988}}.~~

	~~[[Category:Extensions and generalizations of graphs]]~~

Partial correlation - Revision history

31.221.13.140 at 16:28, 17 November 2014

131.251.253.7: /* Semipartial correlation (part correlation) */

en>Terhorst: /* Using matrix inversion */

en>GoingBatty: WPCleaner v1.13 - Repaired 1 link to disambiguation page - (You can help) - Metron