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They also occur in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, and fluid dynamics. The basic hyperbolic functions are: [1]
Applying the Mycielskian repeatedly, starting with the one-edge graph, produces a sequence of graphs M i = μ(M i−1), sometimes called the Mycielski graphs. The first few graphs in this sequence are the graph M 2 = K 2 with two vertices connected by an edge, the cycle graph M 3 = C 5 , and the Grötzsch graph M 4 with 11 vertices and 20 edges.
Differential equations or difference equations on such graphs can be employed to leverage the graph's structure for tasks such as image segmentation (where the vertices represent pixels and the weighted edges encode pixel similarity based on comparisons of Moore neighborhoods or larger windows), data clustering, data classification, or ...
Write the functions without "co" on the three left outer vertices (from top to bottom: sine, tangent, secant) Write the co-functions on the corresponding three right outer vertices (cosine, cotangent, cosecant) Starting at any vertex of the resulting hexagon: The starting vertex equals one over the opposite vertex.
The two edges along the cycle adjacent to any of the vertices are not written down. Let v be the vertices of the graph and describe the Hamiltonian circle along the p vertices by the edge sequence v 0 v 1, v 1 v 2, ...,v p−2 v p−1, v p−1 v 0. Halting at a vertex v i, there is one unique vertex v j at a distance d i joined by a chord with v i,
Several algorithms based on depth-first search compute strongly connected components in linear time.. Kosaraju's algorithm uses two passes of depth-first search. The first, in the original graph, is used to choose the order in which the outer loop of the second depth-first search tests vertices for having been visited already and recursively explores them if not.
A graph with 6 vertices and 7 edges where the vertex number 6 on the far-left is a leaf vertex or a pendant vertex. In discrete mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph ...
The formula was first discovered by Carl Wilhelm Borchardt in 1860, and proved via a determinant. [2] In a short 1889 note, Cayley extended the formula in several directions, by taking into account the degrees of the vertices. [3] Although he referred to Borchardt's original paper, the name "Cayley's formula" became standard in the field.