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| 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.
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| 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''.
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| ==Definition==
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| Formal definition of a <span class="texhtml">Π</span>-voltage graph, for a given group <span class="texhtml">Π</span>:
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| * Begin with a [[digraph (mathematics)|digraph]] ''G''. (The direction is solely for convenience in notation.)
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| * 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'').
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| * 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.
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| * 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.
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| * 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.
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| 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]].
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| ==The derived graph==
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| 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>.
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| 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.
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| 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]].
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| [[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"/>
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| ==Examples==
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| 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>
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| 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>
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| 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>.
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{citation
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| | last1 = Cohen | first1 = Edith
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| | last2 = Megiddo | first2 = Nimrod
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| |authorlink2= Nimrod Megiddo
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| | contribution = Strongly polynomial-time and NC algorithms for detecting cycles in dynamic graphs
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| | doi = 10.1145/73007.73057
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| | pages = 523–534
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| | title = Proc. 21st Annual ACM Symposium on Theory of Computing (STOC '89)
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| | year = 1989}}.
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| *{{citation
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| | last = Gross | first = Jonathan L.
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| | doi = 10.1016/0012-365X(74)90006-5
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| | issue = 3
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| | journal = Discrete Mathematics
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| | pages = 239–246
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| | title = Voltage graphs
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| | volume = 9
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| | year = 1974}}.
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| *{{citation
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| | last1 = Gross | first1 = Jonathan L.
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| | last2 = Tucker | first2 = Thomas W.
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| | doi = 10.1016/0012-365X(77)90131-5
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| | journal = Discrete Mathematics
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| | pages = 273–283
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| | title = Generating all graph coverings by permutation voltage assignments
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| | issue = 3
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| | volume = 18
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| | year = 1977}}.
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| *{{citation
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| | last1 = Gross | first1 = Jonathan L.
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| | last2 = Tucker | first2 = Thomas W.
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| | location = New York
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| | publisher = Wiley
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| | title = Topological Graph Theory
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| | year = 1987}}.
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| *{{citation
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| | last1 = Iwano | first1 = K.
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| | last2 = Steiglitz | first2 = K.
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| | contribution = Testing for cycles in infinite graphs with periodic structure
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| | doi = 10.1145/28395.28401
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| | pages = 46–55
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| | title = Proc. 19th Annual ACM Symposium on Theory of Computing (STOC '87)
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| | year = 1987}}.
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| *{{citation
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| | last1 = Kosaraju | first1 = S. Rao | author1-link = S. Rao Kosaraju
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| | last2 = Sullivan | first2 = Gregory
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| | contribution = Detecting cycles in dynamic graphs in polynomial time
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| | doi = 10.1145/62212.62251
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| | pages = 398–406
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| | title = Proc. 20th Annual ACM Symposium on Theory of Computing (STOC '88)
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| | year = 1988}}.
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| [[Category:Extensions and generalizations of graphs]]
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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.
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