Children: Regular graphs
A Platonic graph is a representation of a Platonic solid, which are the five regular, convex polyhedra that can exist in three-dimensional space. These solids are characterized by having faces that are congruent regular polygons and the same number of faces meeting at each vertex. The five Platonic solids are: 1. Tetrahedron (4 triangular faces) 2. Cube (6 square faces) 3. Octahedron (8 triangular faces) 4.
A Prism graph is a type of polyhedral graph formed by connecting the corresponding vertices of two parallel polytopes, typically two identical polygons. More formally, the prism over a polygon \( P \) can be defined as follows: 1. **Vertices**: The Prism graph has two sets of vertices, each corresponding to the vertices of the polygon \( P \). If \( P \) has \( n \) vertices, then the Prism graph will have \( 2n \) vertices.
A **random regular graph** is a type of graph in which each vertex has the same degree, a property known as **regularity**, and the graph is generated in a random manner. Specifically, a random \( d \)-regular graph is a graph where: 1. **Degree**: Every vertex has exactly \( d \) edges (or connections) to other vertices, meaning it has a degree of \( d \).
A **regular graph** is a type of graph in which every vertex has the same number of edges. This common degree is known as the **degree** of the regular graph. There are two main types of regular graphs: 1. **k-regular**: A graph is k-regular if every vertex has exactly k edges. For example: - A 1-regular graph consists of disjoint edges (pairs of vertices).
The Robertson graph is a specific type of strongly regular graph named after the mathematician Neil Robertson. It is a well-known example in the study of strongly regular graphs, which are a class of graphs characterized by regularity conditions on their vertex connectivity. The Robertson graph has the following properties: - It has 12 vertices. - Each vertex has a degree of 6 (i.e., it is 6-regular). - For any two adjacent vertices, there are exactly 3 common neighbors.
The Robertson–Wegner graph, often discussed in the context of combinatorial graph theory and vertex properties, is a specific type of graph used to illustrate certain structural characteristics in graph theory, particularly for the study of certain properties of graphs such as vertex colorability and independence. ### Key Features 1. **Vertices and Edges**: The Robertson–Wegner graph is illustrated with a specific set of vertices and edges that meet certain combinatorial criteria.
Rook's graph is a type of graph used in graph theory that is derived from the chessboard analogy. Specifically, it represents the possible movements of a rook in chess. To describe Rook's graph more formally: 1. **Vertices**: The vertices of the graph correspond to the squares on a chessboard.
The Schläfli graph is an interesting and well-studied graph in the field of graph theory, particularly in relation to polyhedra and higher-dimensional polytopes. It is defined as the graph whose vertices correspond to the regular polyhedra (in 3D) and regular polytopes (in higher dimensions), and where edges connect pairs of polyhedra that share a common face.
The Shrikhande graph is a specific type of graph in graph theory that is named after the Indian mathematician K. R. Shrikhande. It is a 2-regular graph with 16 vertices and 32 edges, and it is notable for its strong symmetry properties. The Shrikhande graph is defined as follows: - **Vertices**: It has 16 vertices. - **Edges**: It has 32 edges.
A Shuffle-Exchange Network (SEN) is a type of multistage interconnection network used primarily in parallel computing architectures. It is designed to facilitate efficient communication between multiple processors or nodes within a system. The Shuffle-Exchange Network supports operations by efficiently routing data between processors in a way that can help minimize delays and improve communication bandwidth. ### Key Characteristics: 1. **Structure**: The network consists of multiple stages of switches connected in a specific topology.
In graph theory, a **snark** is a specific type of graph that has some interesting properties. Snarks are defined as: 1. **Cubic Graphs**: Snarks are always cubic, meaning every vertex in the graph has a degree of 3. 2. **Not 3-Colorable**: A characteristic feature of snarks is that they cannot be colored with 3 colors without having two adjacent vertices sharing the same color.
A Sudoku graph is a mathematical representation of a Sudoku puzzle using graph theory concepts. In this representation, the elements of the puzzle—such as the numbers in the grid—are mapped to vertices (or nodes) in a graph, and the constraints of Sudoku are represented by edges connecting those vertices. ### Basic Structure of a Sudoku Graph: 1. **Vertices**: Each cell in the Sudoku grid can be represented as a vertex.
A supersingular isogeny graph is a mathematical structure used primarily in number theory and algebraic geometry, particularly in the study of elliptic curves and their isogenies (which are morphisms between elliptic curves that respect the group structure). These graphs have become increasingly important in the field of cryptography, especially in post-quantum cryptographic protocols.
The Suzuki graph is a specific type of graph in the field of graph theory. It is named after mathematician Michio Suzuki, who introduced it in relation to group theory and finite groups. The Suzuki graph is characterized as a strongly regular graph, which means that it has a particular structure based on its vertices and edges.
The Sylvester graph, denoted \( S(n) \), is a specific type of graph that is defined for any positive integer \( n \). It is a vertex-transitive graph that has some intriguing properties, making it interesting in the fields of graph theory and combinatorial design.
The Szekeres snark is a specific type of graph within the field of graph theory, known for its interesting properties. It is a snark, which is a type of non-trivial, cubic graph (meaning each vertex has degree three) that does not have a proper 3-coloring, meaning it cannot be colored with three colors such that no two adjacent vertices share the same color.
A table of simple cubic graphs provides a list of cubic graphs, which are graphs where every vertex has a degree of exactly 3 (i.e., each vertex is connected to exactly three edges). Simple cubic graphs have no loops or multiple edges between the same pair of vertices. These graphs are also known as 3-regular graphs. A common way to organize and present simple cubic graphs is by their number of vertices (usually denoted as \( n \)).
Tietze's graph is a well-known example in graph theory, specifically in the study of planar graphs and their properties. It is a type of graph that is formed by taking a specific arrangement of vertices and edges. The key features of Tietze's graph are: 1. **Vertices and Edges**: Tietze's graph has 12 vertices and 18 edges.
A triangle graph, often referred to in the context of graph theory, can denote different concepts based on context, but generally it refers to a type of graph structure that contains a specific relationship resembling triangles. 1. **Triangle in Graph Theory**: In a general mathematical graph, a triangle is a complete subgraph consisting of three vertices, where each vertex is connected to the other two. This means there are three edges that form a triangle shape.