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).

Luke Cage

Luke Cage (born Carl Lucas and also called Power Man) is a fictional superhero appearing in American comic books published by Marvel Comics. Created by Archie Goodwin and John Romita, Sr., he first appeared in Luke Cage, Hero for Hire #1 (June 1972). Imprisoned for a crime he did not commit, he gains superpowers in the form of unbreakable skin and superhuman strength. The character frequently teams up with fellow superhero Iron Fist, and is married to Jessica Jones, with whom he has a daughter. In 2005, writer Brian Michael Bendis added Luke Cage to the lineup of the New Avengers, and he has appeared in various Avengers titles since.

Actor Mike Colter plays the character in Jessica Jones, a live-action television series set in the Marvel Cinematic Universe, and will headline in his own series, which will premiere in 2016.

Publication history

Luke Cage was created by Archie Goodwin and John Romita, Sr. shortly after Blaxploitation films emerged as a popular new genre. He debuted in his own series, Luke Cage, Hero for Hire, which was initially written by Goodwin and pencilled by George Tuska. Cage's adventures were set in a grungier, more crime-dominated New York City than that inhabited by other Marvel superheroes of the time. The series was retitled Luke Cage, Power Man with issue #17.

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.

Dogū

Dogū (土偶)(meaning "clay figures") are small humanoid and animal figurines made during the late Jōmon period (14,000–400 BC) of prehistoric Japan. Dogū come exclusively from the Jōmon period. By the Yayoi period, which followed the Jōmon period, Dogū were no longer made. There are various styles of Dogū, depending on exhumation area and time period. According to the National Museum of Japanese History, the total number found throughout Japan is approximately 15,000. Dogū were made across all of Japan, except Okinawa. Most of the Dogū have been found in eastern Japan and it is rare to find one in western Japan. The purpose of the Dogū remains unknown and should not be confused with the clay haniwa funerary objects of the Kofun period (250 – 538).

Origins

Some scholars theorize the Dogū acted as effigies of people, that manifested some kind of sympathetic magic. For example, it may have been believed that illnesses could be transferred into the Dogū, then destroyed, clearing the illness, or any other misfortune.

Dog in Chinese mythology

Dogs are an important motif in Chinese mythology. These motifs include a particular dog which accompanies a hero, the dog as one of the twelve totem creatures for which years are named, a dog giving first provision of grain which allowed current agriculture, and claims of having a magical dog as an original ancestor in the case of certain ethnic groups.

Myth versus history

Chinese mythology is those myths found in the geographic area called China, which of course has evolved and changed throughout its history. These include myths in Chinese and other languages, as transmitted by Han Chinese as well as other ethnic groups (of which fifty-six are officially recognized by the current administration of China). (Yang 2005:4)

In the study of historical Chinese culture, many of the stories that have been told regarding characters and events which have been written or told of the distant past have a double tradition: one which tradition which presents a more historicized and one which presents a more mythological version.(Yang 2005: 12-13) This is also true of some accounts related to mythological dogs in China.

D0g

D0g is a fictional character in the Half-Life series of video games, first introduced in Half-Life 2 in 2004. D0g is a hulking robot belonging to Alyx Vance, which was built by Alyx's father Eli to both provide companionship and protect his daughter; Alyx subsequently upgraded the robot into its current form. His character provides comic relief during the series; his battle sequences are often comically exaggerated, such as taking on small enemy squads by throwing cars and trucks at them. His appearance as the first character seen in Half-Life 2: Episode One was influenced by positive fan reception.

Character design

Half-Life 2: Raising the Bar states that D0g's design underwent "relatively few changes" from his very first concept sketch; Valve artists were inspired by "classic movie robots," citing Robby the Robot from Forbidden Planet.

In video games

D0g's role in Half-Life 2 is to aid the player in tutorials—in particular, training in the use of the gravity gun—and lifting heavy objects the player cannot manipulate without the gravity gun.

Dead (disambiguation)

Dead refers to that which has experienced death.

Dead may also refer to:

Arts, entertainment, and media

Music

Artists

Albums

Songs

Television

Chemistry

See also