Voltage graph

In graph-theoretic mathematics, a voltage graph is a directed graph whose edges are labelled invertibly by elements of a 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 called the derived graph of the voltage graph.

Typical choices of the groups used for voltage graphs include the two-element group ℤ₂ (for defining the bipartite double cover of a graph), free groups (for defining the universal cover of a graph), d-dimensional integer lattices ℤ^d (viewed as a group under vector addition, for defining periodic structures in d-dimensional Euclidean space), and finite cyclic groups ℤ_n for n > 2. When Π is a cyclic group, the voltage graph may be called a cyclic-voltage graph.

Formal definition of a Π-voltage graph, for a given group Π:

Note that the voltages of a voltage graph need not satisfy 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 graphs of topological graph theory.

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

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