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In mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. A “graph” in this context refers to a collection of vertices or ‘nodes’ and a collection of edges that connect pairs of vertices. A graph may be undirected, meaning that there is no distinction between the two vertices associated with each edge, or its edges may be directed from one vertex to another; see graph (mathematics) for more detailed definitions and for other variations in the types of graphs that are commonly considered. The graphs studied in graph theory should not be confused with “graphs of functions” and other kinds of graphs.

Please refer to Glossary of graph theory for some basic definitions in graph theory.

This problem was first posed by Francis Guthrie in 1852 and the first written record of this problem is a letter of De Morgan addressed to Hamilton the same year. Many incorrect proofs have been proposed, including those by Cayley, Kempe, and others. The study and the generalization of this problem by Tait, Heawood, Ramsey and Hadwiger has in particular led to the study of the colorings of the graphs embedded on surfaces with arbitrary genus. Tait’s reformulation generated a new class of problems, the factorization problems, particularly studied by Petersen and K?nig. The works of Ramsey on colorations and more specially the results obtained by Turán in 1941 is at the origin of another branch of graph theory, the extremal graph theory.

The autonomous development of topology from 1860 and 1930 fertilized graph theory back through the works of Jordan, Kuratowski and Whitney. Another important factor of common development of graph theory and topology came from the use of the techniques of modern algebra. The first example of such a use comes from the work of the physicist Gustav Kirchhoff, who published in 1845 his Kirchhoff’s circuit laws for calculating the voltage and current in electric circuits.

The introduction of probabilistic methods in graph theory, especially in the study of Erd?s and Rényi of the asymptotic probability of graph connectivity, gave rise to yet another branch, known as random graph theory, which has been a fruitful source of graph-theoretic results.

David Glasgow Farragut (July 5, 1801 – August 14, 1870) was the first senior officer of the United States Navy during the American Civil War. He was the first rear admiral, vice admiral, and full admiral of the Navy. He is remembered in popular culture for his possibly apocryphal order at the Battle of Mobile Bay, usually paraphrased: “Damn the torpedoes, full speed ahead!”.^[1]

Farragut was born to Jorge and Elizabeth Farragut at Lowe’s Ferry on the Holston (now Tennessee) River a few miles south east of Campbell’s Station, near Knoxville, Tennessee, where his family lived. His father operated the ferry and was a cavalry officer in the Tennessee militia. Jorge Farragut (1755 – 1817), a Spanish merchant captain from Minorca, son of Antonio Farragut and Juana Mesquida, had previously joined the American Revolutionary cause after arriving in America in 1776. Jorge Farragut married Elizabeth Shine (b.1765) from North Carolina and moved West to Tennessee after serving in the American Revolution. David’s birth name was James, but it was changed in 1812, following his adoption by future naval Captain David Porter in 1808 (which made him the foster brother of future Civil War Admiral David Dixon Porter and Commodore William D. Porter).

In command of the West Gulf Blockading Squadron, with his flag on the USS Hartford, in April 1862 he ran past Fort Jackson and Fort St. Philip and the Chalmette, Louisiana, batteries to take the city and port of New Orleans, Louisiana, on April 29 that year, a decisive event in the war. His country honored its great sailor after New Orleans by creating for him the rank of rear admiral on July 16, 1862, a rank never before used in the U.S. Navy. (Before this time, the American Navy had resisted the rank of admiral, preferring the term “flag officer”, to separate it from the traditions of the European navies.) Later that year he passed the batteries defending Vicksburg, Mississippi. Farragut had no real success at Vicksburg; one makeshift Confederate ironclad forced his flotilla of 38 ships to withdraw in July 1862.

Russian–Circassian War was the period of hostilities between the Russian Empire and the inhabitants of Circassia during the Russian invasion and occupation of the Circassian region. Circassia (also known as Cherkessia in Russian) was a region in Caucasia which consisted of the coastline and most of the interior of the current territory of Krasnodar Krai and Adygea.^[5] The historical region, now mainly North Ossetia–Alania, was named after the traditional inhabitants, the Circassians, Adyghe or Adiga, along with a number of smaller ethnic groups and tribes.^[6] The Russian–Circassian conflict began with the initial arrival of Russian occupation forces in 1863, and ended with the signing of several Russian loyalty oaths by Circassian leaders on June 2, 1864 (May 21, O.S.).

While the Russian–Circassian War began as an isolated conflict, Russian expansion through the entire region soon brought it into conflict with a number of other nations in what later became known as the Caucasian War, and of which the Russian–Circassian War became a part. Both came to an end with the signing of the loyalty oaths to Russia, and with the total occupation of the region by Russian forces, which involved the mass migration^[1][7][5] of millions of indigenous Circassians to areas of the Ottoman Empire (modern Turkey, Syria, Lebanon, Israel, Jordan and Kosovo),^[5] with some Circassian historians citing that up to 4,000,000 civilians perished during the exodus.^[8]