In
graph theory
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...
, a dipole graph, dipole, bond graph, or linkage, is a
multigraph
In mathematics, and more specifically in graph theory, a multigraph is a graph which is permitted to have multiple edges (also called ''parallel edges''), that is, edges that have the same end nodes. Thus two vertices may be connected by mor ...
consisting of two
vertices connected with a number of
parallel edges. A dipole graph containing edges is called the dipole graph, and is denoted by . The dipole graph is
dual
Dual or Duals may refer to:
Paired/two things
* Dual (mathematics), a notion of paired concepts that mirror one another
** Dual (category theory), a formalization of mathematical duality
*** see more cases in :Duality theories
* Dual number, a nu ...
to the
cycle graph
In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices (at least 3, if the graph is simple) connected in a closed chain. The cycle graph with vertices is called ...
.
The
honeycomb
A honeycomb is a mass of Triangular prismatic honeycomb#Hexagonal prismatic honeycomb, hexagonal prismatic cells built from beeswax by honey bees in their beehive, nests to contain their brood (eggs, larvae, and pupae) and stores of honey and pol ...
as an abstract graph is the maximal abelian
covering graph
In the mathematical discipline of graph theory, a graph is a covering graph of another graph if there is a covering map from the vertex set of to the vertex set of . A covering map is a surjection and a local isomorphism: the neighbourhood of ...
of the dipole graph , while the
diamond crystal as an abstract graph is the maximal abelian covering graph of .
Similarly to the
Platonic graph
In geometric graph theory, a branch of mathematics, a polyhedral graph is the undirected graph formed from the vertices and edges of a convex polyhedron. Alternatively, in purely graph-theoretic terms, the polyhedral graphs are the 3-vertex-con ...
s, the dipole graphs form the skeletons of the
hosohedra
In spherical geometry, an -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices.
A regular -gonal hosohedron has Schläfli symbol with each spherical lune hav ...
. Their duals, the cycle graphs, form the skeletons of the
dihedra
A dihedron (pl. dihedra) is a type of polyhedron, made of two polygon faces which share the same set of ''n'' edges. In three-dimensional Euclidean space, it is degenerate if its faces are flat, while in three-dimensional spherical space, a dih ...
.
References
*
* Jonathan L. Gross and Jay Yellen, 2006. ''Graph Theory and Its Applications, 2nd Ed.'', p. 17. Chapman & Hall/CRC.
*
Sunada T., ''Topological Crystallography, With a View Towards Discrete Geometric Analysis'', Springer, 2013, (Print) 978-4-431-54177-6 (Online)
Extensions and generalizations of graphs
Parametric families of graphs
Regular graphs
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