SUMMARY / RELATED TOPICS

In graph theory, a regular graph is a graph where each vertex has the same number of neighbors. A regular directed graph must satisfy the stronger condition that the indegree and outdegree of each vertex are equal to each other. A regular graph with vertices of degree k is called a k‑regular graph or regular graph of degree k. From the handshaking lemma, a regular graph of odd degree will contain an number of vertices. Regular graphs of degree at most 2 are easy to classify: A 0-regular graph consists of disconnected vertices, a 1-regular graph consists of disconnected edges, a 2-regular graph consists of a disjoint union of cycles and infinite chains. A 3-regular graph is known as a cubic graph. A regular graph is a regular graph where every adjacent pair of vertices has the same number l of neighbors in common, every non-adjacent pair of vertices has the same number n of neighbors in common; the smallest graphs that are regular but not regular are the cycle graph and the circulant graph on 6 vertices.

The complete graph K m is regular for any m. A theorem by Nash-Williams says that every k‑regular graph on 2k + 1 vertices has a Hamiltonian cycle, it is well known that the necessary and sufficient conditions for a k regular graph of order n to exist are that n ≥ k + 1 and that n k is even. In such case it is easy to construct regular graphs by considering appropriate parameters for circulant graphs. Let A be the adjacency matrix of a graph; the graph is regular if and only if j = is an eigenvector of A. Its eigenvalue will be the constant degree of the graph. Eigenvectors corresponding to other eigenvalues are orthogonal to j, so for such eigenvectors v =, we have ∑ i = 1 n v i = 0. A regular graph of degree k is connected if and only; the "only if" direction is a consequence of the Perron–Frobenius theorem. There is a criterion for regular and connected graphs: a graph is connected and regular if and only if the matrix of ones J, with J i j = 1, is in the adjacency algebra of the graph. Let G be a k-regular graph with diameter D and eigenvalues of adjacency matrix k = λ 0 > λ 1 ≥ ⋯ ≥ λ n − 1.

If G is not bipartite D ≤ log ⁡ log ⁡ + 1. Fast algorithms exist to enumerate, up to isomorphism, all regular graphs with a given degree and number of vertices. Random regular graph Strongly regular graph Moore graph Cage graph Highly irregular graph Weisstein, Eric W. "Regular Graph". MathWorld. Weisstein, Eric W. "Strongly Regular Graph". MathWorld. GenReg software and data by Markus Meringer. Nash-Williams, Valency Sequences which force graphs to have Hamiltonian Circuits, University of Waterloo Research Report, Ontario: University of Waterloo

Mooswald is a district in the western part of Freiburg im Breisgau, which consists of the two districts Mooswald West and Mooswald East. In the North East, Mooswald borders the district of Brühl, with its airfield, the University campus of the technical faculty and the exhibition centre. Mooswald further borders the district of Landwasser in the North West, the district of Stühlinger in the East and the district of Betzenhausen with the Seepark in the South West. Mooswald is separated from Brühl by the Breisgau S-Bahn, from Landwasser by the Westrandstraße and from Stühlinger by the railway tracks of the freight railway. In 1932, the city council of Freiburg decided on the construction of a suburb settlement at the former western border of the city, the Mooswald, after the government of the German Reich decided on an unemployment programme in 1931, in times of high unemployment; the houses were built independently by the settlers and could be purchased as leasehold estates. After 1933 the project was continued as "Nationalsozialistisches Siedlungswerk".

In 1934, a settler union was established. The simple houses had neither electricity nor gas nor a connection to the sewage system at the beginning. At the beginning the water supply was achieved with pump wells on the properties. Big gardens enabled the self-sufficiency of the settlers with the cultivation of potatoes and vegetables and allowed the keeping of small animals. In 1938, the Catholic Church Holy Family was built in this region. During the air raid on Freiburg in November 1944, 80% of the district was destroyed. In 1945, active reconstruction work with simple means started. In 1948, the station "Im Wolfswinkel" was established at the railway line to Breisach. In 1952, today's civic association Mooswald e. V. was founded. The city of Freiburg established a landfill site in the nearby "Wolfswinkel". In 1953, the Evangelic St. Mark Church was built as an emergency church; the district received a primary school with the Mooswald school in 1955/56, a community house was completed. After the end of World War II, the preservation of the western part of the Mooswald was a permanent topic for the inhabitants of the district of Mooswald, since the forest became affected more and more by the city's expansion to the west.

There are two churches in this district: the "Holy Family" Catholic St. Mark's Church. Moreover, the Jewish cemetery of Freiburg is situated there. There are various schools in the district of Mooswald: the Paul Hindemith primary school, the Wentzinger schools, the Mooswaldschule, which nowadays is a school for educational aids. A meeting centre for young and old is the Fritz Hüttinger Haus, managed by the civic association Mooswald and whose rooms are used for cultural events, language courses, or as a meeting point for the youth; the western Mooswälder forests and the Seepark lying in the southwest of Freiburg in the district of Betzenhausen provide relaxation opportunities. The Breisacher Hof is a housing estate in the former Gallwitz barracks, on the move. Early in 2017, the city announced that the football ground there is to be developed into 60 funded tenement flats by 2020. In 2013, the "Westarkaden", a shopping centre, was completed. More tenement flats are under construction. Quite striking is a radio tower of the German Telecom near the Paduaallee.

The construction work of the tram line 4 to "Messe" began in 2013. On 11 December 2015, the first section of the tramway towards the temporary terminus Technische Fakultät became operational. By 2018, the last kilometer of tracks in the direction of "Messe" should be finished; the Breisgau-S-Bahn Freiburg-Breisach serves the stop Neue Messe/Universität. The main road in Freiburg is Elsässerstraße on which the Freiburger Verkehrs AG line 10 bus operates. Four-lane highways in the west of the city and the east of the city make the district accessible for cars. Bund für Umwelt und Naturschutz: "Abholzung Mooswald Freiburg?: Gefährdete Natur in der "Green" City " Jüdischer Friedhof

Nodosauridae is a family of ankylosaurian dinosaurs, from the Late Jurassic to the Late Cretaceous period of what are now North America, Asia and Antarctica. Nodosaurids, like their close relatives the ankylosaurids, were armored dinosaurs adorned with rows of bony armor nodules and spines, which were covered in keratin sheaths. All nodosaurids, like other ankylosaurians, were medium-sized to large built, herbivorous dinosaurs, possessing small, leaf-shaped teeth. Unlike ankylosaurids, nodosaurids lacked mace-like tail clubs. Many nodosaurids had spikes projecting outward from their shoulders. One well-preserved nodosaurid "mummy", known as the Suncor nodosaur, preserved a nearly complete set of armor in life position, as well as the keratin covering and mineralized remains of the underlying skin, which indicate the animal had red and white camouflage; the family Nodosauridae was erected by Othniel Charles Marsh in 1890, anchored on the genus Nodosaurus. The clade Nodosauridae was first defined by Paul Sereno in 1998 as "all ankylosaurs closer to Panoplosaurus than to Ankylosaurus," a definition followed by Vickaryous, Teresa Maryańska, Weishampel in 2004.

Vickaryous et al. considered two genera of nodosaurids to be of uncertain placement: Struthiosaurus and Animantarx, considered the most primitive member of the Nodosauridae to be Cedarpelta. The cladogram below follows the most resolved topology from a 2011 analysis by paleontologists Richard S. Thompson, Jolyon C. Parish, Susannah C. R. Maidment and Paul M. Barrett; the placement of Polacanthinae follows its original definition by Kenneth Carpenter in 2001. The near simultaneous appearance of nodosaurids in both North America and Europe is worthy of consideration. Europelta is the oldest nodosaurid from Europe, it is derived from the lower Albian Escucha Formation; the oldest western North American nodosaurid is Sauropelta, from the lower Albian Little Sheep Mudstone Member of the Cloverly Formation, at an age of 108.5±0.2 million years. Eastern North American fossils seem older. Teeth of Priconodon crassus from the Arundel Clay of the Potomac Group of Maryland, which dates near the Aptian–Albian boundary.

The Propanoplosaurus hatchling from the base of the underlying Patuxent Formation, dating to the upper Aptian, is the oldest known nodosaurid. Polacanthids are known from pre-Aptian fauna from both North America; the timing of the appearance of nodosaurids on both continents indicates that the origins of the clade preceded the isolation of North America and Europe, pushing the group's date of evolution back to at least the "middle" Aptian. The separation of Nodosauridae into European Struthiosaurinae and North American Nodosaurinae by the end of the Aptian provides a revised date for the isolation of the continents from each other by rising sealevels. Below is a table showing the age difference between continents. North American nodosaurids are teal, European nodosaurids are green, European polacanthids are blue, North American polacanthids are brown. Other nodosaurids or polacanthids are black; this table supports the observations by Kirkland et al.. James Kirkland et al. considers Mymoorapelta, Hylaeosaurus, Polacanthus and Gastonia to be Polacanthids, outside of Nodosauridae.

Timeline of ankylosaur research Carpenter, K.. "Phylogenetic analysis of the Ankylosauria." In Carpenter, K. 2001: The Armored Dinosaurs. Indiana University Press, Bloomington & Indianapolis, 2001, pp. xv-526 Osi, Attila. Hungarosaurus tormai, a new ankylosaur from the Upper Cretaceous of Hungary. Journal of Vertebrate Paleontology 25:370-383, June 2003. Alberta oilsands discovery of 2011

The Jackson Homestead, located at 527 Washington Street, in the village of Newton Corner, in Newton, Massachusetts, is an historic house that served as a station on the Underground Railroad before the Civil War. It was built in 1809 in the Federal style by Timothy Jackson on his family's farm, his son William Jackson lived in it from 1820 until his death. William Jackson was an abolitionist and was active in politics on the local and national levels and served in the United States Congress from 1833 to 1837; the home was occupied by his family until 1932. In 1949 it was given to the city of Newton and in 1950 the Newton History Museum was established there. In 1646, Edward Jackson bought a 500-acre farm which covered much of Newton Newtonville. In about 1670, he built a house on the side of east side of Smelt Brook, on the site of the present Homestead; the south-facing house was a saltbox: two stories in front and one at the back 22 by 18 feet. There were two kitchens and a bedroom on the lower floor, two chambers on the second, two further bedrooms in the attic.

The house, never painted, either inside or out, took water from a well sheltered by a large elm tree. 17 feet were added to the house either by Edward's son Sebas or by his grandson, Joseph. As was not uncommon, there were always two generations living in the house, the addition was to provide for a second family. In 1809, Major Timothy Jackson decided to replace the old homestead with "a fine house for the times", he designed and helped build a large, two-family house, hoping that his youngest son, would move into it if he married. The house was a great improvement on the original, featuring such improvements as an inside well, a laundry and ell, magnificent fireplaces with huge mantelpieces, a great staircase and airy, small private bedchambers on the second floor, with a large, useful garrett extending on top of the entire house. Timothy Jackson had built. After Timothy Jr's death, his estate was divided among his sons. William Jackson struck deals with his brothers, buying the shares from Edmund and George, made a "division" with Francis and Stephen, according to which they took the land to the south of Washington street and he received the homestead and everything to the north.

Adjustments were made to the house, most notably the creation and splitting of bedrooms, to accommodate the large family, the house was painted yellow with the same cream trim and green shutters the Homestead has today, central heating would be installed in the 1830s. By the 1900s, the Jackson Homestead began using municipal water rather from their indoor well. Under the direction of Louise Keith, William's granddaughter, major renovations were carried out. Modern plumbing was installed in the bathroom and kitchen, all of which were connected to the city sewers. Wooden ceiling beams were reinforced with iron braces and seventy-six new panels of glass were installed; the interior of the house was repainted, the exterior of the house was painted white for the first time. In 1949, Frances Paine, gave the homestead to the City of Newton for educational and other public purposes; the Homestead became the CIty Museum, a number of changes have been made, most notably the remodelling of the ell in 1966.