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Sylvester Graph
The Sylvester graph is the unique distance-regular graph with intersection array \ . It is a subgraph of the Hoffman–Singleton graph In the mathematical field of graph theory, the Hoffman–Singleton graph is a 7- regular undirected graph with 50 vertices and 175 edges. It is the unique strongly regular graph with parameters (50,7,0,1). It was constructed by Alan Hoffman an .... References {{Reflist External links A.E. Brouwer's website: the Sylvester graph Individual graphs Regular graphs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distance Regular Graph
In the mathematical field of graph theory, a distance-regular graph is a regular graph such that for any two vertices and , the number of vertices at distance from and at distance from depends only upon , , and the distance between and . Every distance-transitive graph is distance-regular. Indeed, distance-regular graphs were introduced as a combinatorial generalization of distance-transitive graphs, having the numerical regularity properties of the latter without necessarily having a large automorphism group. Intersection arrays It turns out that a graph G of diameter d is distance-regular if and only if there is an array of integers \ such that for all 1 \leq j \leq d , b_j gives the number of neighbours of u at distance j+1 from v and c_j gives the number of neighbours of u at distance j - 1 from v for any pair of vertices u and v at distance j on G . The array of integers characterizing a distance-regular graph is known as its intersection array. Cos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Graph
In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex exactly once. A Hamiltonian path that starts and ends at adjacent vertices can be completed by adding one more edge to form a Hamiltonian cycle, and removing any edge from a Hamiltonian cycle produces a Hamiltonian path. Determining whether such paths and cycles exist in graphs (the Hamiltonian path problem and Hamiltonian cycle problem) are NP-complete. Hamiltonian paths and cycles are named after William Rowan Hamilton who invented the icosian game, now also known as ''Hamilton's puzzle'', which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron. Hamilton solved this problem using the icosian calculus, an algebraic structure based on roots of unity with many similarities to the quaternions (also invented by Hami ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hoffman–Singleton Graph
In the mathematical field of graph theory, the Hoffman–Singleton graph is a 7- regular undirected graph with 50 vertices and 175 edges. It is the unique strongly regular graph with parameters (50,7,0,1). It was constructed by Alan Hoffman and Robert Singleton while trying to classify all Moore graphs, and is the highest-order Moore graph known to exist. Since it is a Moore graph where each vertex has degree 7, and the girth is 5, it is a (7,5)-cage. Construction Here are two constructions of the Hoffman–Singleton graph. Construction from pentagons and pentagrams Take five pentagons ''Ph'' and five pentagrams ''Qi'' . Join vertex ''j'' of ''Ph'' to vertex ''h''·''i''+''j'' of ''Qi''. (All indices are modulo 5.) Construction from PG(3,2) Take a Fano plane on seven elements, such as and apply all 2520 even permutations on the 7-set ''abcdefg''. Canonicalize each such Fano plane (e.g. by reducing to lexicographic order) and discard duplicates. Exactly 15 Fano planes remai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Individual Graphs
An individual is that which exists as a distinct entity. Individuality (or self-hood) is the state or quality of being an individual; particularly (in the case of humans) of being a person unique from other people and possessing one's own needs or goals, rights and responsibilities. The concept of an individual features in diverse fields, including biology, law, and philosophy. Etymology From the 15th century and earlier (and also today within the fields of statistics and metaphysics) ''individual'' meant " indivisible", typically describing any numerically singular thing, but sometimes meaning "a person". From the 17th century on, ''individual'' has indicated separateness, as in individualism. Law Although individuality and individualism are commonly considered to mature with age/time and experience/wealth, a sane adult human being is usually considered by the state as an "individual person" in law, even if the person denies individual culpability ("I followed instruct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
