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(A degenerate triangle, whose vertices are collinear, has internal angles of 0° and 180°; whether such a shape counts as a triangle is a matter of convention. [ citation needed ] ) The conditions for three angles α {\displaystyle \alpha } , β {\displaystyle \beta } , and γ {\displaystyle \gamma } , each of them between 0° and 180°, to be ...
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 ...
In graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree and outdegree of each internal vertex are equal to each other. [1]
The complete graph on n vertices is denoted by K n.Some sources claim that the letter K in this notation stands for the German word komplett, [4] but the German name for a complete graph, vollständiger Graph, does not contain the letter K, and other sources state that the notation honors the contributions of Kazimierz Kuratowski to graph theory.
A level-5 approximation to a Sierpiński triangle obtained by shading the first 2 5 (32) levels of a Pascal's triangle white if the binomial coefficient is even and black otherwise If one takes Pascal's triangle with 2 n {\displaystyle 2^{n}} rows and colors the even numbers white, and the odd numbers black, the result is an approximation to ...
The neighbourhood of a vertex v in a graph G is the subgraph of G induced by all vertices adjacent to v, i.e., the graph composed of the vertices adjacent to v and all edges connecting vertices adjacent to v. The neighbourhood is often denoted or (when the graph is unambiguous) . The same neighbourhood notation may also be used ...
Sperner's lemma states that, if a big triangle is subdivided into smaller triangles meeting edge-to-edge, and the vertices are labeled with three colors so that only two of the colors are used along each edge of the big triangle, then at least one of the smaller triangles has vertices of all three colors; it has applications in fixed-point ...
If the set of vertices of a simplex is , …,, then considering the vertices as vectors, the centroid is C = 1 n + 1 ∑ i = 0 n v i . {\displaystyle C={\frac {1}{n+1}}\sum _{i=0}^{n}v_{i}.} The geometric centroid coincides with the center of mass if the mass is uniformly distributed over the whole simplex, or concentrated at the vertices as n ...
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