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In graph theory, the degree diameter problem is the problem of finding the largest possible graph G of diameter k such that the largest degree of any of the vertices in G is at most d. The size of G is bounded above by the Moore bound. In general, the largest degree-diameter graphs are much smaller in size than the Moore bound. Let n d, k be the maximum possible number of vertices for a graph with degree at most d and diameter k. N d, k ≤ M d, k, where M d, k is the Moore bound: M d, k = { 1 + d k − 1 d − 2 if d > 2 2 k + 1 if d = 2 This bound is attained for few graphs, thus the study moves to how close there exist graphs to the Moore bound. For asymptotic behaviour note that M d, k = d k + O. Define the parameter μ k = lim inf d → ∞ n d, k d k, it is conjectured that μ k = 1 for all k. It is known that μ 1 = μ 2 = μ 3 = μ 5 = 1 and that μ 4 ≥ 1 / 4. For the general case it is known that μ k ≥ 1.6 k. Thus, although it is conjectured that μ k = 1, it is still an open question whether it is in fact exponential.

Cage Table of degree diameter graphs Table of vertex-symmetric degree diameter digraphs Maximum degree-and-diameter-bounded subgraph problem Bannai, E.. "On Moore graphs", J. Fac. Sci. Univ. Tokyo Ser. A, 20: 191–208, MR 0323615Hoffman, Alan J.. "There is no irregular Moore graph", American Mathematical Monthly, Mathematical Association of America, 75: 42–43, doi:10.2307/2315106, JSTOR 2315106, MR 0225679Miller, Mirka.

"Runaway Train" is a power ballad by American rock band Soul Asylum. Its music video is notable for featuring images of missing people, most of them involving young children and adolescent teenagers. Although lead singer Dave Pirner states that the lyrics described his experience of depression, it was released in June 1993 as the fourth single from their 1992 album, Grave Dancers Union and became a success around the world. In mid 1993, it reached number five on the Billboard Hot 100, number two on the U. S. Top 40 climbed to the top position on the Canadian RPM Top Singles chart, it was certified gold by the Recording Industry Association of America and sold 600,000 copies in the United States. Worldwide, the song was successful, reaching number two in New Zealand, Norway and Switzerland and peaking within the top five of several other European countries; the song helped bring their album, Grave Dancers Union, to a multi-platinum level and won a Grammy Award for Best Rock Song in 1994. The music video, directed by Tony Kaye, received heavy airplay on VH1 during its duration.

Several versions of the video were made. The video for the United States version begins with a fade to a black screen with a big, white blocked text reading: "There are over one million youth lost on the streets of America", while the UK version begins with "100,000 youth are lost on the streets of Britain"; the next scene shows a drawing of an adolescent girl, a Dave Pirner voice-over saying that the drawing is by a girl who had run away more than 110 times. The scene was omitted when the video was shown, a common practice when videos had additional footage before or after the song. After Pirner spoke, the video continued with various shots of the band playing the song, Dave singing. Three concrete scenes are shown interspersed among the other images of the video. During the first verse, a child is shown witnessing his grandfather beating, killing his grandmother, running from their house in fear. During the second verse, a young teenage girl is pimped as a prostitute, is purchased by the aforementioned abuser.

She is dragged into a van by a gang, afterwards she is picked up by paramedics and taken to the hospital, after getting beaten up. During the coda of the song, a small baby is snatched from his stroller by an older woman, with his mother running after the kidnapper's car. Throughout the music video, various images of children running, or appearing with injuries from abuse, are shown. During the choruses, pictures of missing children would appear on the screen. After each picture was shown, their full name would appear in large capital letters on the screen, along with the year they had been "missing since...". After the video, in an ending not shown, Pirner says in front of the camera, "If you've seen one of these kids, or you are one of them, please call this number," with the following screen showing a number one could contact. MTV cut this part out because they did not want to have the video confused with being a public service announcement. VH1 shows the UK version in its full length. There were three original versions of the video in the United States, totaling 36 missing children shown.

The children shown varied with the location of the broadcast. According to Kaye, 26 missing children were found after being featured in the video. In 2006, guitarist Dan Murphy stated in an interview with Pasadena Weekly that some of the cases featured in the video had ended in tragedy: "Some weren't the best scenarios. I met a fireman on the East Coast whose daughter was in the end of the video, he'd been in a bitter custody battle with his wife over her", Murphy said. "It turned out the girl was killed and buried in her backyard by her mother. On tour, another girl told us laughingly'You ruined my life' because she saw herself on the video at her boyfriend's house and it led her being forced back into a bad home situation."The UK version of the video featured Vicky Hamilton and Dinah McNicol, who each went missing in 1991. Their remains were found in 2007 at a house in Margate. Peter Tobin has since been convicted of both murders. Featured in the UK version was Mark Bartley, a runaway who went missing in 1992.

He was recognised in the video by a man who knew Bartley was staying in the tenant's house below them, but was unaware of his missing status. By the time the police arrived and the man he was living with were gone, it is unknown. Curtis Huntzinger, featured in the US video, was located deceased in 2008, his convicted killer, Stephen Daniel Hash, pleaded guilty to manslaughter and, in 2009, was sentenced to 11 years in Folsom State Prison. The last image in all three U. S. versions of the song is Thomas Dean Gibson, who disappeared from Glendale, Oregon, in 1991 at the age of 2. He is still missing as of 2019, age-progressed photos of him at age 19 and age 21 were released in 2009 and 2012 by the National Center for Missing and Exploited Children, his father, Larry Gibson, a former deputy sheriff, was convicted of second degree manslaughter for accidentally shooting his son to death when he shot at a stray cat in his front yard though no remains were found. He steadfastly denies killing his son and has worked on finding him since being released from prison in 1996.

The version shown in Australia showed a number of young backpacking tourists whose families were looking for them. Many of those shown in the Australian version were confirmed victims of serial killer Ivan Milat, arrested in 1994 not long after the Australian film clip was released. Featured in the video, but still missing in 2019 were Christopher Kerze, Martha Dunn, Andrea Dur

The Waverly Gallery is a play by Kenneth Lonergan. It is considered a "memory play"; the show, first produced Off-Broadway in 2000, follows a grandson watching his grandmother die from Alzheimer's disease. The play was a finalist for the Pulitzer Prize for Drama in 2001; the play opened Off-Broadway at the Promenade Theater on March 22, 2000 and closed on May 21, 2000. Directed by Scott Ellis, the play starred Eileen Heckart as Gladys Green and Josh Hamilton as Daniel; the play premiered at the Williamstown Theatre Festival, running from August 11, 1999 to August 22, 1999. Joanne Woodward filled in for an ailing Eileen Heckart in the final four performances; the play premiered on Broadway at the John Golden Theatre on September 25, 2018 in previews on October 25. The cast included: Elaine May as Gladys Lucas Hedges as Daniel Joan Allen as Ellen Michael Cera as Don David Cromer as HowardThe revival was directed by Lila Neugebauer; the play closed on January 2019 after 109 performances. Gladys Green owns a small art gallery in Greenwich Village.

She is in her 80s and showing signs of Alzheimer's disease. Don, a young artist, arrives for a showing of his work; the landlord wants to replace it with a restaurant. How her family – daughter Ellen, son-in-law Howard and grandson Daniel – deals with her decline is told by the grandson. Charles Isherwood in Variety said, "The life trauma being depicted has an inherent pathos, in Lonergan's hands, no small amount of comic potential, and yet, while Lonergran mines his subject with delicacy and wit, he runs out of dramatic ore well before the evening's end."Ben Brantley in The New York Times called the play a "finely observed story of the predations of old age... isn't so much a proper play as an essayistic memoir given dramatic form. It is nonetheless theatrical. Mr. Lonergan... has one of the keenest ears of any working playwright.... is often funny." The Waverly Gallery was a finalist for the Pulitzer Prize for Drama in 2001. Awards and Nominations: The 2000 Drama Desk Awards - Outstanding Actress - Winner The 1999–2000 Obie Award - Performance - Winner The 2000 Lucille Lortel Award - Outstanding Actress - Winner The 2000 Drama League Award - Distinguished Performance - Winner The 2000 Outer Critics Circle Award, John Glassner Award - Nominated 2019 Drama Desk Awards Outstanding Revival of a Play - Winner Outstanding Actress in a Play - Winner2019 Tony Awards Nominations Tony Award for Best Revival of a Play Tony Award for Best Actress in a Play - Winner Internet Broadway Database

Magnus Alexander Maximilian Freiherr von Braun was a German civil servant and conservative politician whose career spanned the German Empire, World War I and the Weimar Republic. He served as the Federal Minister of Nutrition and Agriculture from 1 June 1932 to 28 January 1933. Magnus von Braun was born at his family's manor of Neucken, an estate the von Brauns had owned since 1803, near Pr. Eylau in East Prussia to Maximilian von Braun and Eleonore, he studied law at the Universities of Göttingen and Königsberg and joined the Prussian civil service in 1905, at first at the Department of Trade and Commerce in Berlin. With his father's death in 1918, he inherited the title of Freiherr. Between 1911 and 1915 he was the district chief executive of the Kreis Wirsitz, he returned to Berlin in 1915 after being given an office with the Department of the Interior. In September 1917 Braun became the first chief press officer for the Reich Chancellery and the head of the political department of the military administration of Vilnius.

He became the Stadthauptmann for the Baltic town of Daugavpils in 1918. Following the conclusion of World War I, Braun was appointed Commissarial Police President of Stettin in 1919. Braun worked again at the Department of Interior and became the President of the Governorate of Gumbinnen, he was dismissed from the civil service after the Kapp Putsch in 1920 for his role in the coup. Braun returned to his family's manor in East Prussia and was active in several agricultural organisations like the Raiffeisen cooperative. In 1930 he became the Vice President of the Reichsverband der Landwirtschaftlichen Genossenschaften. On 1 June 1932 he was appointed Weimar Germany's Minister of Nutrition and Agriculture and Reichskommissar for Eastern Aid in the cabinet of Chancellor Franz von Papen, a position he kept under Chancellor Kurt von Schleicher until 28 January 1933. After the Nazis came to power on 30 January 1933, Braun moved to his manor in Silesia, which after World War II became part of Poland and Braun was expelled to Western Germany in 1946.

Braun followed his son Wernher to the United States in 1947, but returned to Germany in 1952, where he died in 1972 at Oberaudorf. Braun married Emmy von Quistorp on 12 July 1910, they had three sons: Sigismund von Braun, diplomat Wernher von Braun, rocket scientist Magnus von Braun, industrial manager Genealogy wiki

Nob Nation is the title of a series of topical comedy pieces broadcast since November 2006 on two national radio stations in Ireland, RTÉ 2fm and RTÉ Radio 1. The programmes are broadcast Monday through Saturday each week. Nob Nation is produced by Oliver Callan; the series features 100 characters, including BIFFO, Dame Enda, Garret FitzGerald, others based on people from Irish and international public life involved in politics and entertainment. All voice acting is performed by Callan. Two CDs of Nob Nation sketches have been released, in 2007 and a follow-up in 2008. In 2011, RTÉ paid defamation damages to an 84-year-old gentleman after Nob Nation inaccurately claimed he was running a brothel. Broadcast on The Derek Mooney Show on RTÉ Radio 1 and subsequently on the 2fm Breakfast Show, the sketches were moved to just after the 10:00 news on The Gerry Ryan Show on RTÉ 2fm in January 2007. Nob Nation was repeated at night on Damien Farrelly's The Frequency on 2FM. One comedy sketch was selected and was featured on Playback on Radio One on Saturday mornings and on The Saturday Show on 2fm on Saturday afternoons.