Cage (graph theory)

In the mathematical area of graph theory, a cage is a regular graph that has as few vertices as possible for its girth.

Formally, an (r,g)-graph is defined to be a graph in which each vertex has exactly r neighbors, and in which the shortest cycle has length exactly g. It is known that an (r,g)-graph exists for any combination of r ≥ 2 and g ≥ 3. An (r,g)-cage is an (r,g)-graph with the fewest possible number of vertices, among all (r,g)-graphs.

If a Moore graph exists with degree r and girth g, it must be a cage. Moreover, the bounds on the sizes of Moore graphs generalize to cages: any cage with odd girth g must have at least

vertices, and any cage with even girth g must have at least

vertices. Any (r,g)-graph with exactly this many vertices is by definition a Moore graph and therefore automatically a cage.

There may exist multiple cages for a given combination of r and g. For instance there are three nonisomorphic (3,10)-cages, each with 70 vertices : the Balaban 10-cage, the Harries graph and the Harries–Wong graph. But there is only one (3,11)-cage : the Balaban 11-cage (with 112 vertices).

Robertson graph

In the mathematical field of graph theory, the Robertson graph or (4,5)-cage, is a 4-regular undirected graph with 19 vertices and 38 edges named after Neil Robertson.

The Robertson graph is the unique (4,5)-cage graph and was discovered by Robertson in 1964. As a cage graph, it is the smallest 4-regular graph with girth 5.

It has chromatic number 3, chromatic index 5, diameter 3, radius 3 and is both 4-vertex-connected and 4-edge-connected.

The Robertson graph is also a Hamiltonian graph which possesses 5,376 distinct directed Hamiltonian cycles.

Algebraic properties

The Robertson graph is not a vertex-transitive graph and its full automorphism group is isomorphic to the dihedral group of order 24, the group of symmetries of a regular dodecagon, including both rotations and reflections.

The characteristic polynomial of the Robertson graph is

Gallery

The Robertson graph as drawn in the original publication.

The Robertson graph as drawn in the original publication.

The chromatic number of the Robertson graph is 3.

Cage9

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Cage9 is a Panamanian/American alternative rock band, formed in Panama City in 1993. The group was founded by Evan Rodaniche (guitar/vocals) and is currently based out of Los Angeles, California.

Cage9 has been direct openers for Ozzy Osbourne and Bad Religion, as well as shared the stage with notable acts like Queensrÿche, Candlebox, Seether, Sevendust, Hellyeah, Filter, In This Moment, Tesla, Karnivool, Buckcherry, MxPx, Powerman 5000, Dokken, Quiet Riot, Lynch Mob, Hinder, Boy Hits Car and Smile Empty Soul.

The band's song "My Doppelgänger (Doble Opuesto)," was released in English as well as Spanish and received full rotation on MTV Latino. The song "Hearts & Stars" is on the soundtrack to the video game MX vs. ATV: Untamed. Cage9 recorded a version of the theme song to The Dukes of Hazzard which was used in the trailer for the 2005 movie. The band also recorded music for a WWE promotion for Stone Cold Steve Austin and composed the Kamen Rider: Dragon Knight theme song, called "Let's Ride". On March 3, 2006, Cage9 made an appearance on G4's Attack of the Show!.

Home directory

A home directory is a file system directory on a multi-user operating system containing files for a given user of the system. The specifics of the home directory (such as its name and location) is defined by the operating system involved; for example, Windows systems between 2000 and 2003 keep home directories in a folder called Documents and Settings.

Description

A user's home directory is intended to contain that user's files; including text documents, music, pictures or videos, etc. It may also include their configuration files of preferred settings for any software they have used there and might have tailored to their liking: web browser bookmarks, favorite desktop wallpaper and themes, passwords to any external services accessed via a given software, etc. The user can install executable software in this directory, but it will only be available to users with permission to this directory. The home directory can be organized further with the use of sub-directories.

The content of a user's home directory is protected by file system permissions, and by default is accessible to all authenticated users and administrators. Any other user that has been granted administrator privileges has authority to access any protected location on the filesystem including other users home directories.

Home (Deep Blue Something album)

Home is the second album by alternative rock band Deep Blue Something. It was originally released by RainMaker Records in 1994 and re-released on Interscope in 1995.

Track listing

All songs written by Todd Pipes, except where noted.

B-Sides:

Charts

Personnel

Band Members

Home (Stephanie Mills album)

Home is a studio album by Stephanie Mills. It was released June 26, 1989 on MCA Records.

Track listing

Charts

References

External links