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In graph theory, a simplicial vertex v {\displaystyle v} is a vertex whose closed neighborhood N G [ v ] {\displaystyle N_{G}[v]} in a graph G {\displaystyle
Simplicial_vertex
The link in a simplicial complex is a generalization of the neighborhood of a vertex in a graph. The link of a vertex encodes information about the local
Link_(simplicial_complex)
Fundamental unit of which graphs are formed
denoted 𝛿−(v); a source vertex is a vertex with indegree zero, while a sink vertex is a vertex with outdegree zero. A simplicial vertex is one whose closed
Vertex_(graph_theory)
Point where two or more curves, lines, or edges meet
complexes such as simplicial complexes are its zero-dimensional faces. A polygon vertex xi of a simple polygon P is a principal polygon vertex if the diagonal
Vertex_(geometry)
Mathematical object
of an abstract simplicial complex Δ and the vertex set V(Δ) ⊆ S of Δ: for the purposes of defining a category of abstract simplicial complexes, the elements
Abstract_simplicial_complex
Mathematical construction used in homotopy theory
arrows, and directed loops that connect a vertex to itself are also allowed. Unlike directed multigraphs, simplicial sets may also contain higher simplices
Simplicial_set
graph is weakly recursively simplicial if it has a simplicial vertex and the subgraph after removing a simplicial vertex and some edges (possibly none)
Moral_graph
sibling of a vertex v is a vertex which has the same parent vertex as v. simplicial vertex A simplicial vertex is a vertex whose closed neighborhood forms
Glossary_of_graph_theory
Type of mathematical set
(x)} . Two simplices and their closure. A vertex and its star. A vertex and its link. Let K be a simplicial complex and let S be a collection of simplices
Simplicial_complex
Concept in algebraic topology
In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of
Simplicial_homology
Set of polygons to define the surface of a 3D model
ways, using different methods to store the vertex, edge and face data. These include: vertex-vertex face-vertex winged-edge half-edge quad-edge Each representation
Polygon_mesh
neighborhood of each vertex (i.e. the set of simplices that contain that point as a vertex) is homeomorphic to a n-dimensional ball. A simplicial manifold is also
Simplicial_manifold
Shape made by slicing off a corner of a polytope
cubes and one octahedron around the other edges. Simplicial link - an abstract concept related to vertex figure. List of regular polytopes Coxeter, H. et
Vertex_figure
A simplicial map (also called simplicial mapping) is a function between two simplicial complexes, with the property that the images of the vertices of
Simplicial_map
Representation of mathematical space
correspond to vertex sets of simplices in S {\displaystyle {\mathcal {S}}} . A natural question is if vice versa, any abstract simplicial complex corresponds
Triangulation_(topology)
Tiling of n-dimensional space
In geometry, the simplicial honeycomb (or n-simplex honeycomb) is a dimensional infinite series of honeycombs, based on the A ~ n {\displaystyle {\tilde
Simplicial_honeycomb
Algorithm for linear programming
it is that it operates on simplicial cones, and these become proper simplices with an additional constraint. The simplicial cones in question are the
Simplex_algorithm
Construction for n-dimensional noise functions
An implementation typically involves four steps: coordinate skewing, simplicial subdivision, gradient selection, and kernel summation. An input coordinate
Simplex_noise
Multi-dimensional generalization of triangle
simplices to form a simplicial complex. The geometric simplex and simplicial complex should not be confused with the abstract simplicial complex, in which
Simplex
Complex in algebraic topology
topology and topological data analysis, the Čech complex is an abstract simplicial complex constructed from a point cloud in any metric space which is meant
Čech_complex
Subdivision of the plane by lines
arrangement is said to be simple when at most two lines cross at each vertex, and simplicial when all cells are triangles (including the unbounded cells, as
Arrangement_of_lines
Continuous mappings can be approximated by ones that are piecewise simple
In mathematics, the simplicial approximation theorem is a foundational result for algebraic topology, guaranteeing that continuous mappings can be (by
Simplicial approximation theorem
Simplicial_approximation_theorem
Two pentagonal pyramids fused base-to-base
four four-connected simplicial well-covered graphs. It is also one of the six connected graphs in which its neighborhood of every vertex is a cycle of length
Pentagonal_bipyramid
Subgraph induced by all nodes linked to a given node of a graph
neighborhood problem Vertex figure, a related concept in polyhedra Link (simplicial complex), a generalization of the neighborhood to simplicial complexes Hell
Neighbourhood_(graph_theory)
omnitruncated simplex facets. The vertex figure for each is an irregular n-simplex. The facets of an omnitruncated simplicial honeycomb are called permutahedra
Omnitruncated simplicial honeycomb
Omnitruncated_simplicial_honeycomb
Algebraic structure associated with a topological space
which make the task easier. The simplicial homology groups Hn(X) of a simplicial complex X are defined using the simplicial chain complex C(X), with Cn(X)
Homology_(mathematics)
Abstraction useful in the construction and triangulation of topological spaces
In mathematics, a Δ-set, often called a Δ-complex or a semi-simplicial set, is a combinatorial object that is useful in the construction and triangulation
Delta_set
Topological data
removing from the simplicial complex the highest order simplexes until the simplicial complex is empty. We then need to label each vertex from 1 to | V |
Simplex_tree
Topics referred to by the same term
loan in California and Nevada, US Simplicial link, a set of simplices "surrounding" a given vertex in a simplicial complex Link (knot theory), a collection
Link
Newest Vertex Bisection is an algorithmic method to locally refine triangulations. It is widely used in computational science, numerical simulation, and
Newest_vertex_bisection
N-dimensional polytope with vertices adjacent to N facets
(also d facets). The vertex figure of a simple d-polytope is a (d – 1)-simplex. Simple polytopes are topologically dual to simplicial polytopes. The family
Simple_polytope
Fourth letter in the Greek alphabet
the position of which is variant between isomeric forms. A simplex, simplicial complex, or convex hull. In chemistry, the addition of heat in a reaction
Delta_(letter)
Abstract simplicial complex describing a graph's cliques
hypergraphs to the language of simplicial complexes. The barycentric subdivision of any cell complex C is a flag complex having one vertex per cell of C. A collection
Clique_complex
Concept in algebraic topology
1-skeleton is also known as the vertex-edge graph of the polytope. The above definition of the skeleton of a simplicial complex is a particular case of
N-skeleton
Concept in graph theory
only if each edge belongs to the closed neighborhood of a simplicial vertex. These simplicial vertices correspond to the maximal elements of the underlying
Bound_graph
vertex arrangement: Hypercubic honeycomb Alternated hypercubic honeycomb Quarter hypercubic honeycomb Simplectic honeycomb Omnitruncated simplicial honeycomb
Cyclotruncated simplicial honeycomb
Cyclotruncated_simplicial_honeycomb
In mathematics, a dendroidal set is a generalization of simplicial sets introduced by Moerdijk & Weiss (2007). They have the same relation to (colored
Dendroidal_set
and Gerald Reisner in the early 1970s. Given an abstract simplicial complex Δ on the vertex set {x1,...,xn} and a field k, the corresponding Stanley–Reisner
Stanley–Reisner_ring
Solid with eight equal triangular faces
a four-connected simplicial well-covered graph. It is also one of the six connected graphs in which the neighborhood of every vertex is a cycle of length
Regular_octahedron
Part of the mathematical subject of group theory
analyzing the algebraic structure of groups acting by automorphisms on simplicial trees. The theory relates group actions on trees with decomposing groups
Bass–Serre_theory
Simplicial set constructed from the objects and morphisms of a small category
small category C is a simplicial set constructed from the objects and morphisms of C. The geometric realization of this simplicial set is a topological
Nerve_(category_theory)
Map between simplicial sets with lifting property
part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category structure on simplicial sets and are therefore
Kan_fibration
independence complex of an undirected graph G, denoted by I(G), is an abstract simplicial complex (that is, a family of finite sets closed under the operation of
Independence_complex
Shape with three sides
as the simplex, and the polytopes with triangular facets known as the simplicial polytopes. Each triangle has many special points inside it, on its edges
Triangle
Graph with equal-size maximal independent sets
minimum vertex cover. The independence complex of a graph G is the simplicial complex having a simplex for each independent set in G. A simplicial complex
Well-covered_graph
In polytope theory, the edge graph (also known as vertex-edge graph or just graph) of a polytope is a combinatorial graph whose vertices and edges correspond
Graph_of_a_polytope
Generalized manifold
consists of: for each vertex i of X ', a simplicial complex Li' endowed with a rigid simplicial action of a finite group Γi. a simplicial map φi of Li' onto
Orbifold
Branch of geometry that studies combinatorial properties and constructive methods
illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory
Discrete_geometry
Formalism in general relativity
In general relativity, Regge calculus is a formalism for producing simplicial approximations of spacetimes that are solutions to the Einstein field equation
Regge_calculus
Polyhedron with 6 faces
regular hexahedron with all its faces square, and three squares around each vertex. There are seven topologically distinct convex hexahedra, one of which exists
Hexahedron
Area of discrete mathematics
spaces. The graph in a topology is a set of simplexes that is called the simplicial one-dimensional complex. This subarea studies the embedding (or imbedding)
Graph_theory
Two tetrahedra joined by one face
of its triangular faces with any type, the triangular bipyramid is a simplicial polyhedron like other infinitely many bipyramids. A right bipyramid is
Triangular_bipyramid
Adjacent subset of an undirected graph
is an abstract simplicial complex X(G) with a simplex for every clique in G A simplex graph is an undirected graph κ(G) with a vertex for every clique
Clique_(graph_theory)
Vertices connected in pairs by edges
and v and to be incident on them. A vertex may belong to no edge, in which case it is not joined to any other vertex and is called isolated. When an edge
Graph_(discrete_mathematics)
graph, together with a family of monomorphisms of the edge groups into the vertex groups. There is a unique group, called the fundamental group, canonically
Graph_of_groups
Independent set in a graph
independent set in a graph, in which each vertex has a different color. Formally, let G = (V, E) be a graph, and suppose vertex set V is partitioned into m subsets
Rainbow-independent_set
Polytope made by turning a polytope's facets into pyramids
In particular, the Kleetope of any three-dimensional polyhedron is a simplicial polyhedron, a polyhedron in which all facets are triangles. Kleetopes
Kleetope
Generalization of graph theory
C ) ∈ E {\displaystyle (D,C)\in E} is called an edge or hyperedge; the vertex subset D {\displaystyle D} is known as its tail or domain, and C {\displaystyle
Hypergraph
Non-orientable surface with one edge
come from an abstract simplicial complex, because all three triangles share the same three vertices, while abstract simplicial complexes require each
Möbius_strip
Combinitorics of Polyhedra
connectivity and diameter (number of steps needed to reach any vertex from any other vertex). Additionally, many computer scientists use the phrase “polyhedral
Polyhedral_combinatorics
Tree graph with one central node and leaves of length 1
Starlike tree - a tree with at most one vertex of degree larger than 2; a subdivision of a star Star (simplicial complex) - a generalization of the concept
Star_(graph_theory)
Topological invariant in mathematics
(When only triangular faces are used, they are two-dimensional finite simplicial complexes.) In general, for any finite CW-complex, the Euler characteristic
Euler_characteristic
Convex hull of a finite set of points in a Euclidean space
e. as a spherical tiling. A convex polytope can be decomposed into a simplicial complex, or union of simplices, satisfying certain properties. Given a
Convex_polytope
Convex polyhedron projected from hypercube
zonohedron corresponds in this way to a simplicial arrangement, one in which each face is a triangle. Simplicial arrangements of great circles correspond
Zonohedron
Discrete (i.e., incremental) version of infinitesimal calculus
A simplicial map f {\displaystyle f} from a simplicial complex S {\displaystyle S} to another T {\displaystyle T} is a function from the vertex set
Discrete_calculus
Shape in hyperbolic geometry
cannot be realized as ideal polyhedra. If a simplicial polyhedron (one with all faces triangles) has all vertex degrees between four and six (inclusive)
Ideal_polyhedron
Mathematical group of the homotopy classes of loops in a topological space
covering space of a finite connected simplicial complex X {\displaystyle X} can also be described directly as a simplicial complex using edge-paths. Its vertices
Fundamental_group
Mathematical structure
simplicial complex formed by all (n − 1)! simplices with a given common vertex in the analogous tessellation in En−2. Each building is a simplicial complex
Building_(mathematics)
Theorem on Hamiltonian graphs
2-vertex-connected if it does not have an articulation vertex, a vertex whose deletion would leave the remaining graph disconnected. Not every 2-vertex-connected
Fleischner's_theorem
dimensional cubical complex is locally CAT(0) iff all its vertex links are flag simplicial. complexes. Cubical complexes have a wide range of applications
Cubical_complex
Subdivision of a planar object into triangles
many simplices in T {\displaystyle T} . That is, it is a locally finite simplicial complex that covers the entire space. A point-set triangulation, i.e.
Triangulation_(geometry)
Line constructed from a triangle
Euler line. A simplicial polytope is a polytope whose facets are all simplices (plural of simplex). For example, every polygon is a simplicial polytope. The
Euler_line
Graph made from vertices and edges of a convex polyhedron
of a simple polyhedron if it is cubic (every vertex has three edges), and it is the graph of a simplicial polyhedron if it is a maximal planar graph. For
Polyhedral_graph
Topological manifold whose homology coincides with that of a sphere
not a PL manifold. In other words, this gives an example of a finite simplicial complex that is a topological manifold but not a PL manifold. (It is not
Homology_sphere
Operation in topology
{\displaystyle B} are any abstract simplicial complexes, then their join is an abstract simplicial complex defined as follows: The vertex set V ( A ⋆ B ) {\displaystyle
Join_(topology)
Topological space arising from a usual graph
{\displaystyle y} . That is, as topological spaces, graphs are exactly the simplicial 1-complexes and also exactly the one-dimensional CW complexes. Thus, in
Graph_(topology)
Mathematical category
topos a pro-simplicial set (up to homotopy). (It's better to consider it in Ho(pro-SS); see Edwards) Using this inverse system of simplicial sets one may
Topos
the significance of stacked polytopes is that, among all d-dimensional simplicial polytopes with a given number of vertices, the stacked polytopes have
Stacked_polytope
Mathematical object in topological graph theory
complex Δ m , n {\displaystyle \Delta _{m,n}} is the abstract simplicial complex with vertex set [ m ] × [ n ] {\displaystyle [m]\times [n]} that contains
Chessboard_complex
Generalizations in graph theory
{\mathcal {M}}(H)} . It is a simplicial complex on the edges of H, whose elements are all the matchings on H. For each vertex y in Y, let Vy be set of edges
Hall-type theorems for hypergraphs
Hall-type_theorems_for_hypergraphs
Planar surface that forms part of the boundary of a solid object
concept that generalizes some earlier types of polyhedra is the notion of a simplicial complex. More generally, there is the notion of a polytopal complex. An
Face_(geometry)
Branch of the mathematical field of graph theory
undirected graph we may associate an abstract simplicial complex C with a single-element set per vertex and a two-element set per edge. The geometric
Topological_graph_theory
"holes" in the graph. It is a special case of a simplicial homology, as a graph is a special case of a simplicial complex. Since a finite graph is a 1-complex
Graph_homology
that one gets a simplicial structure on SPn(X). Furthermore, SPn(X) is also a subsimplex of SPn+1(X) if the basepoint e ∈ X is a vertex, meaning that SP(X)
Symmetric_product_(topology)
Regular tiling of the plane
of the rows. The vertex arrangement of the triangular tiling is called an A2 lattice. It is the 2-dimensional case of a simplicial honeycomb. The A*
Triangular_tiling
2 means that X is a non-branching simplicial complex. Condition 3 means that X is a strongly connected simplicial complex. If we require Condition 2
Pseudomanifold
Generalization of depth-first search trees
depth-first search and connecting each vertex (other than the starting vertex of the search) to the earlier vertex from which it was discovered. The tree
Trémaux_tree
g-conjecture on the possible numbers of faces of different dimensions in a simplicial sphere (also Grünbaum conjecture, several conjectures of Kühnel) (Karim
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Combinatorial representation of a graph on an orientable surface
representation and processing, in geometrical modeling. This model is related to simplicial complexes and to combinatorial topology. A combinatorial map is a boundary
Combinatorial_map
The degree-Rips bifiltration filters each simplicial complex in the Rips filtration by the degree of each vertex in the graph isomorphic to the 1-skeleton
Degree-Rips_bifiltration
topological graph cross a finite number of times, no edge passes through a vertex different from its endpoints, and no two edges touch each other (without
Topological_graph
Class of undirected graphs defined from systems of sets
Ilan; Rabinovich, Yuri (2015), On Connectivity of the Facet Graphs of Simplicial Complexes, arXiv:1502.02232, Bibcode:2015arXiv150202232N. Rispoli, Fred
Johnson_graph
Flat-sided three-dimensional shape
of faces, topological classification by Euler characteristic, duality, vertex figures, surface area, volume, interior lines, Dehn invariant, and symmetry
Polyhedron
{\displaystyle n} vertices, adjoin 1 to the Cartesian coordinates of each vertex, to obtain a ( d + 1 ) {\displaystyle (d+1)} -dimensional column vector
Gale_diagram
category D. Set, the category of (small) sets. sSet, the category of simplicial sets. "weak" instead of "strict" is given the default status; e.g., "n-category"
Glossary_of_category_theory
parameter. Often, the Vietoris–Rips filtration is used to create a discrete, simplicial model on point cloud data embedded in an ambient metric space. The Vietoris–Rips
Vietoris–Rips_filtration
Mathematical theory
L)&\cong 0\\\end{aligned}}} Let X {\displaystyle X} be an abstract simplicial complex on a vertex set V {\displaystyle V} of size n {\displaystyle n} . The Alexander
Alexander_duality
Hypothetical approach to quantum gravity with emergent spacetime
by a discrete time variable t. Each space slice is approximated by a simplicial manifold composed by regular (d − 1)-dimensional simplices and the connection
Causal dynamical triangulation
Causal_dynamical_triangulation
Theorem in group theory
{\displaystyle G} admits a nontrivial (that is, without a global fixed vertex) action on a simplicial tree with finite edge-stabilizers and without edge-inversions
Stallings theorem about ends of groups
Stallings_theorem_about_ends_of_groups
Undirected graph with 11 nodes and 27 edges
same graph had already been given as an example of a non-Hamiltonian simplicial polyhedron by Branko Grünbaum in 1967. The Goldner–Harary graph is a planar
Goldner–Harary_graph
SIMPLICIAL VERTEX
SIMPLICIAL VERTEX
Girl/Female
Indian
Simplicity and purity
Girl/Female
Indian
Simplicity and purity
Boy/Male
Hindu, Indian
More Polite; Simplicity
Boy/Male
Indian, Punjabi, Sikh
Love for Simplicity
Boy/Male
Indian, Punjabi, Sikh
Victory of Simplicity
Girl/Female
Hindu, Indian, Tamil
One with Simplicity; Special Person of All Beings
Girl/Female
Tamil
Hitansi | ஹிதாஂஸீ
Simplicity and purity
Hitansi | ஹிதாஂஸீ
Girl/Female
Bengali, Hindu, Indian, Kannada, Marathi, Sanskrit, Tamil, Telugu
Goddess Laxmi; Prosperity; Simplicity; Lovable; Affectionate; Wealthy; Fortunate
Girl/Female
Greek Latin Spanish
Pastoral simplicity and happiness.
Girl/Female
Tamil
Hitanshi | ஹிதாஂஷீÂ
Simplicity and purity
Hitanshi | ஹிதாஂஷீÂ
Girl/Female
Assamese, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Tamil, Telugu
Virtuous Woman; Simplicity
SIMPLICIAL VERTEX
SIMPLICIAL VERTEX
Boy/Male
Hindu
Lover
Girl/Female
Bengali, Gujarati, Hindu, Indian
Image of Shiva; Name of Flower; God Shiva's Wife
Girl/Female
Tamil
Well heard, A good reputation, Very famous
Female
Greek
(ΧαÏικλώ) Greek name KHARIKLO means "graceful spinner." In mythology, this is the name of the nymph wife of Kheiron (Latin Chiron) the centaur.
Female
English
Anglicized form of Hebrew Yediydah, JEDIDAH means "friend" or "beloved." In the bible, this is the name of the mother of king Josiah.
Boy/Male
Hindu, Indian
Lotus Flower
Girl/Female
Hindu, Indian
One who Gives Our Secrets
Boy/Male
Tamil
Mind
Boy/Male
Tamil
Towards the fire
Girl/Female
Indian
Green, Name of a Goddess
SIMPLICIAL VERTEX
SIMPLICIAL VERTEX
SIMPLICIAL VERTEX
SIMPLICIAL VERTEX
SIMPLICIAL VERTEX
n.
Freedom from subtlety or abstruseness; clearness; as, the simplicity of a doctrine; the simplicity of an explanation or a demonstration.
n.
Artlessness of mind; freedom from cunning or duplicity; lack of acuteness and sagacity.
n.
Simplicity; silliness.
n.
Simplicity or plainness, bordering on weakness or silliness; artlessness; ingenuousness.
n.
The state of being elementary; original simplicity; uncompounded state.
n.
One who is simple.
n.
The state or quality of being childish; simplicity; harmlessness; weakness of intellect.
n.
Native simplicity; unaffected plainness or ingenuousness; artlessness.
n.
Simplicity.
n.
Want of wisdom; unwise conduct or action; folly; simplicity; ignorance.
n.
Plainness; freedom from adornment; severe simplicity.
n.
The quality or state of being simple; simplicity.
n.
Weakness of intellect; silliness; folly.
n.
The quality or state of being simple, unmixed, or uncompounded; as, the simplicity of metals or of earths.
n.
Coarseness; simplicity; want of refinement; as, the homeliness of manners, or language.
n.
The quality or state of being not complex, or of consisting of few parts; as, the simplicity of a machine.
n.
Freedom from artificial ornament, pretentious style, or luxury; plainness; as, simplicity of dress, of style, or of language; simplicity of diet; simplicity of life.
n.
The quality or state of being rustic; rustic manners; rudeness; simplicity; artlessness.
n.
The quality of being artless, or void of art or guile; simplicity; sincerity.
n.
Absence of simplicity; artfulness.