Children: Regular graphs
The Frankl–Rödl graph is a specific type of undirected graph that is characterized by certain properties and can be defined based on combinatorial structures. It is named after mathematicians Victor Frankl and Hans rödl, who studied properties related to graph theory and combinatorics.
The Frucht graph is a specific type of graph in graph theory, notable for being the smallest cubic (3-regular) graph that is also Hamiltonian and non-vertex-transitive. It has 12 vertices and 18 edges, and is a useful example in the study of graph properties. Key characteristics of the Frucht graph include: 1. **Cubic Graph**: All vertices in the Frucht graph have degree 3.
The Generalized Petersen graph is a family of graphs that generalize the structure of the well-known Petersen graph. These graphs are denoted as \( GP(n, k) \), where \( n \) and \( k \) are positive integers. The Generalized Petersen graph is defined using two parameters: - \( n \): the number of vertices in the outer cycle (which is a simple cycle graph with \( n \) vertices).
A Gewirtz graph is a specific type of graph in graph theory that is defined based on a particular recursive construction process. Named after the mathematician Herbert Gewirtz, it can be constructed by starting with a base graph and performing a series of operations that generate new edges and vertices based on certain rules. The most commonly associated features of Gewirtz graphs include the following: 1. **Recursive Construction**: Gewirtz graphs can be built incrementally.
The Gosset graph, also known as the 7-dimensional hypercube graph, is a specific geometric structure in graph theory and is associated with the symmetrical properties of certain polytopes. It can be thought of as a high-dimensional extension of more familiar concepts, similar to how the cube relates to the square. The Gosset graph has a total of 7 vertices, and each vertex is connected to 3 other vertices.
A Grassmann graph, also known as a Grassmannian graph, is a concept from the field of combinatorial geometry and algebraic geometry that is closely related to Grassmannians. Grassmannians are spaces that parameterize all k-dimensional linear subspaces of an n-dimensional vector space. The vertices of a Grassmann graph correspond to the k-dimensional subspaces of a vector space, and the edges represent the relationships between these subspaces.
A **Gray graph**, often referred to in the context of Gray codes, is a graph that represents the relationships between different binary codes generated by changing one bit at a time. In mathematical terms, a Gray graph typically represents the vertices and edges formed by these codes. ### Gray Code A Gray code is a binary numeral system where two successive values differ in only one bit.
The Hall–Janko graph is a well-known graph in the field of graph theory and combinatorial design. It is named after mathematicians Philip Hall and J. M. Janko. The graph has the following characteristics: 1. **Vertices and Edges**: The Hall–Janko graph consists of 100 vertices and 300 edges. 2. **Regular**: It is a strongly regular graph with parameters \((100, 30, 0, 12)\).
The Halved Cube Graph, often denoted as \( Q_n' \), is a specific graph that is derived from the n-dimensional hypercube graph \( Q_n \). The hypercube graph \( Q_n \) consists of vertices representing all binary strings of length \( n \), where two vertices are connected by an edge if their corresponding binary strings differ by exactly one bit.
A Hamming graph, denoted as \( H(n, d) \), is a type of graph that represents the relationships between binary strings of a certain length and the Hamming distance between them. Specifically, the Hamming graph \( H(n, d) \) is defined as follows: - **Vertices**: Each vertex corresponds to a binary string of length \( n \).
The Harries graph, also known as a Hassler graph, is a specific type of graph in the field of graph theory. In such graphs, vertices are connected through edges in a manner that satisfies particular conditions. Harries graphs are often studied for their properties in relation to connectivity, chromatic number, and other characteristics. However, it is worth noting that there are many specific types of graphs, and "Harries graph" may not be a widely recognized term in all contexts.
The Harries–Wong graph is a specific type of graph used in combinatorial mathematics and graph theory. It is particularly known for being a counterexample to certain conjectures in graph theory, especially related to the properties of extremal graphs—graphs that maximize or minimize a particular property under specified conditions. The graph is constructed using a specific method and has been researched for its unique characteristics in the context of colorings, coverings, and other properties.
The Heawood graph is a specific type of graph in graph theory that serves as an important example in various areas, including topology and combinatorics. It is named after the mathematician Percy John Heawood, who studied it in the context of map coloring problems. Here are some key features of the Heawood graph: 1. **Structure**: The Heawood graph is a bipartite graph with 14 vertices and 21 edges.
The Higman–Sims graph is a highly symmetric, 22-vertex graph that arises in the context of group theory and combinatorial design. It is named after mathematicians Graham Higman and Charles Sims, who studied its properties in relation to the Higman–Sims group, a specific group in group theory. Here are some important characteristics of the Higman–Sims graph: 1. **Vertices and Edges**: The graph has 22 vertices and 57 edges.
The Hoffman graph is a specific undirected graph that is notable in the study of graph theory. It is defined as a graph on 12 vertices and 18 edges. The graph is often used for various theoretical discussions, particularly in the context of properties of graphs such as symmetry, cliques, and its relation to other types of graphs.
The Hoffman–Singleton graph is a highly symmetric, 7-regular graph with 50 vertices. It is named after mathematicians Alan Hoffman and R. R. Singleton, who discovered it in the context of coding theory. Here are some key properties of the Hoffman–Singleton graph: 1. **Vertices and Edges**: It has 50 vertices and 175 edges. Each vertex has a degree of 7, meaning that each vertex is connected to 7 other vertices.
A Holt graph is a type of graphical representation used to visualize relationships between nodes in a network, specifically focusing on the outcomes of a Holt-style forecasting method in time series analysis. The Holt method involves two smoothing parameters: one for the level of the series and another for the trend. The graphs typically highlight how forecasts evolve over time, illustrating both the historical data and the predictions based on the model.
A Horton graph is a specific type of graph named after the mathematician and computer scientist, C. V. Horton. It is particularly known for its application in the study of hierarchical structures and networks, typically in relation to social sciences, biological systems, or computer science.
A **hypercube graph**, often denoted as \( Q_n \), is a graph that represents the relationships between the vertices of an \( n \)-dimensional hypercube. The vertices of the hypercube correspond to the binary strings of length \( n \), and there is an edge between two vertices if the corresponding binary strings differ in exactly one bit position.
A Johnson graph, denoted as \( J(n, k) \), is a type of vertex-transitive graph that represents the relationships between the \( k \)-element subsets of an \( n \)-element set. Specifically, the vertices of a Johnson graph are the \( k \)-element subsets of a set with \( n \) elements, and there is an edge between two vertices (subsets) if their intersection has exactly \( k-1 \) elements.