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Petersen Family
In graph theory, the Petersen family is a set of seven undirected graphs that includes the Petersen graph and the complete graph . The Petersen family is named after Danish mathematician Julius Petersen, the namesake of the Petersen graph. Any of the graphs in the Petersen family can be transformed into any other graph in the family by Δ-Y or Y-Δ transforms, operations in which a triangle is replaced by a degree-three vertex or vice versa. These seven graphs form the forbidden minors for linklessly embeddable graphs, graphs that can be embedded into three-dimensional space in such a way that no two cycles in the graph are linked. They are also among the forbidden minors for the YΔY-reducible graphs. Definition The form of Δ-Y and Y-Δ transforms used to define the Petersen family is as follows: *If a graph contains a vertex with exactly three neighbors, then the Y-Δ transform of at is the graph formed by removing from and adding edges between each pair of its ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minor (graph Theory)
Minor may refer to: * Minor (law), a person under the age of certain legal activities. ** A person who has not reached the age of majority * Academic minor, a secondary field of study in undergraduate education Music theory * Minor chord ** Barbershop seventh chord or minor seventh chord * Minor interval * Minor key * Minor scale Mathematics * Minor (graph theory), the relation of one graph to another given certain conditions * Minor (linear algebra), the determinant of a certain submatrix People * Charles Minor (1835–1903), American college administrator * Charles A. Minor (21st-century), Liberian diplomat * Dan Minor (1909–1982), American jazz trombonist * Dave Minor (1922–1998), American basketball player * James T. Minor, US academic administrator and sociologist * Jerry Minor (born 1969), American actor, comedian and writer * Kyle Minor (born 1976), American writer * Mike Minor (actor) (born 1940), American actor * Mike Minor (baseball) (born 1987), Ameri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Apex Graph
In graph theory, a branch of mathematics, an apex graph is a graph that can be made planar by the removal of a single vertex. The deleted vertex is called an apex of the graph. It is ''an'' apex, not ''the'' apex because an apex graph may have more than one apex; for example, in the minimal nonplanar graphs or , every vertex is an apex. The apex graphs include graphs that are themselves planar, in which case again every vertex is an apex. The null graph is also counted as an apex graph even though it has no vertex to remove. Apex graphs are closed under the operation of taking minors and play a role in several other aspects of graph minor theory: linkless embedding, Hadwiger's conjecture,. YΔY-reducible graphs, and relations between treewidth and graph diameter. Characterization and recognition Apex graphs are closed under the operation of taking minors: contracting any edge, or removing any edge or vertex, leads to another apex graph. For, if is an apex graph with ape ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star (graph Theory)
In graph theory, a star is the complete bipartite graph a tree (graph theory), tree with one internal node and leaves (but no internal nodes and leaves when ). Alternatively, some authors define to be the tree of order (graph theory), order with maximum diameter (graph theory), diameter 2; in which case a star of has leaves. A star with 3 edges is called a claw. The star is Edge-graceful labeling, edge-graceful when is even and not when is odd. It is an edge-transitive matchstick graph, and has diameter 2 (when ), Girth (graph theory), girth ∞ (it has no cycles), chromatic index , and chromatic number 2 (when ). Additionally, the star has large automorphism group, namely, the symmetric group on letters. Stars may also be described as the only connected graphs in which at most one vertex has degree (graph theory), degree greater than one. Relation to other graph families Claws are notable in the definition of claw-free graphs, graphs that do not have any claw as a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Horst Sachs
Horst Sachs (27 March 1927 – 25 April 2016) was a German mathematician, an expert in graph theory, a recipient of the Euler Medal (2000). He earned the degree of Doctor of Science (Dr. rer. nat.) from the Martin-Luther-Universität Halle-Wittenberg in 1958. Following his retirement in 1992, he was professor emeritus at the Institute of Mathematics of the Technische Universität Ilmenau. His encyclopedic book in spectral graph theory, ''Spectra of Graphs. Theory and Applications'' (with Dragos Cvetković and Michael Doob) has several editions and was translated in several languages.Review by P. Rowlinson (1996), ''Proceedings of the Edinburgh Mathematical Society (Series 2)'' 39: 188–189, . Two theorems in graph theory bear his name. One of them relates the coefficients of the characteristic polynomial of a graph to certain structural features of the graph. Another one is a simple relation between the characteristic polynomials of a graph and its line graph In the ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycle (graph Theory)
In graph theory, a cycle in a graph is a non-empty trail in which only the first and last vertices are equal. A directed cycle in a directed graph is a non-empty directed trail in which only the first and last vertices are equal. A graph without cycles is called an ''acyclic graph''. A directed graph without directed cycles is called a '' directed acyclic graph''. A connected graph without cycles is called a ''tree''. Definitions Circuit and cycle * A circuit is a non-empty trail in which the first and last vertices are equal (''closed trail''). : Let be a graph. A circuit is a non-empty trail with a vertex sequence . * A cycle or simple circuit is a circuit in which only the first and last vertices are equal. Directed circuit and directed cycle * A directed circuit is a non-empty directed trail in which the first and last vertices are equal (''closed directed trail''). : Let be a directed graph. A directed circuit is a non-empty directed trail with a vertex seq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension, including the three-dimensional space and the ''Euclidean plane'' (dimension two). The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of '' proving'' all properties of the space as theorems, by starting from a few fundamental properties, called ''postulates'', which either were considered as evident (for example, there is exactly one straight line passing through two points), or seemed impossible to prove (paral ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
