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Prism graph
In the mathematical field of graph theory, a prism graph is a graph that has one of the prisms as its skeleton.
The individual graphs may be named after the associated solid:
Although geometrically the star polygons also form the faces of a different sequence of (self-intersecting and non-convex) prismatic polyhedra, the graphs of these star prisms are isomorphic to the prism graphs, and do not form a separate sequence of graphs.
Prism graphs are examples of generalized Petersen graphs, with parameters GP(n,1). They may also be constructed as the Cartesian product of a cycle graph with a single edge.
As with many vertex-transitive graphs, the prism graphs may also be constructed as Cayley graphs. The order-n dihedral group is the group of symmetries of a regular n-gon in the plane; it acts on the n-gon by rotations and reflections. It can be generated by two elements, a rotation by an angle of 2π/n and a single reflection, and its Cayley graph with this generating set is the prism graph. Abstractly, the group has the presentation (where r is a rotation and f is a reflection or flip) and the Cayley graph has r and f (or r, r−1, and f) as its generators.
The n-gonal prism graphs with odd values of n may be constructed as circulant graphs . However, this construction does not work for even values of n.
The graph of an n-gonal prism has 2n vertices and 3n edges. They are regular, cubic graphs. Since the prism has symmetries taking each vertex to each other vertex, the prism graphs are vertex-transitive graphs. As polyhedral graphs, they are also 3-vertex-connected planar graphs. Every prism graph has a Hamiltonian cycle. even sided prism graphs are bipartite graphs.
Among all biconnected cubic graphs, the prism graphs have within a constant factor of the largest possible number of 1-factorizations. A 1-factorization is a partition of the edge set of the graph into three perfect matchings, or equivalently an edge coloring of the graph with three colors. Every biconnected n-vertex cubic graph has O(2n/2) 1-factorizations, and the prism graphs have Ω(2n/2) 1-factorizations.
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Prism graph
In the mathematical field of graph theory, a prism graph is a graph that has one of the prisms as its skeleton.
The individual graphs may be named after the associated solid:
Although geometrically the star polygons also form the faces of a different sequence of (self-intersecting and non-convex) prismatic polyhedra, the graphs of these star prisms are isomorphic to the prism graphs, and do not form a separate sequence of graphs.
Prism graphs are examples of generalized Petersen graphs, with parameters GP(n,1). They may also be constructed as the Cartesian product of a cycle graph with a single edge.
As with many vertex-transitive graphs, the prism graphs may also be constructed as Cayley graphs. The order-n dihedral group is the group of symmetries of a regular n-gon in the plane; it acts on the n-gon by rotations and reflections. It can be generated by two elements, a rotation by an angle of 2π/n and a single reflection, and its Cayley graph with this generating set is the prism graph. Abstractly, the group has the presentation (where r is a rotation and f is a reflection or flip) and the Cayley graph has r and f (or r, r−1, and f) as its generators.
The n-gonal prism graphs with odd values of n may be constructed as circulant graphs . However, this construction does not work for even values of n.
The graph of an n-gonal prism has 2n vertices and 3n edges. They are regular, cubic graphs. Since the prism has symmetries taking each vertex to each other vertex, the prism graphs are vertex-transitive graphs. As polyhedral graphs, they are also 3-vertex-connected planar graphs. Every prism graph has a Hamiltonian cycle. even sided prism graphs are bipartite graphs.
Among all biconnected cubic graphs, the prism graphs have within a constant factor of the largest possible number of 1-factorizations. A 1-factorization is a partition of the edge set of the graph into three perfect matchings, or equivalently an edge coloring of the graph with three colors. Every biconnected n-vertex cubic graph has O(2n/2) 1-factorizations, and the prism graphs have Ω(2n/2) 1-factorizations.