Algebraic graph theory

Algebraic graph theory is a branch of mathematics that studies graphs through algebraic methods and concepts. It combines ideas from both graph theory, which is the study of graphs—objects consisting of vertices (or nodes) connected by edges—and various areas of algebra, particularly linear algebra and group theory.

Cayley graphs are a type of graph used in group theory to represent the structure of a group in a visual and geometric way. Named after the mathematician Arthur Cayley, these graphs provide insight into the group's properties, including symmetries and relationships among its elements. ### Definition: A Cayley graph is constructed from a group \( G \) and a generating set \( S \) of that group.

A **Cayley graph** is a graphical representation of a group that illustrates the structure and properties of the group in terms of its generators. Named after the mathematician Arthur Cayley, this type of graph provides a way to visualize how elements of a group can be combined (or multiplied) to produce other elements.

In the context of group theory, the growth rate refers to a concept that assesses how the number of elements in the finite index subgroups of a group grows with respect to their index. More specifically, the growth rate can describe how the size of the balls in the Cayley graph of a group increases as the radius of the ball grows, which is tied to the group’s algebraic structure and properties.

The term "superstrong approximation" doesn't refer to a widely recognized concept in mainstream scientific literature or mathematics as of my last knowledge update in October 2023. It’s possible that it could refer to an advanced concept in a specialized field, or it might be a term that has emerged more recently or is used informally in specific contexts.

A **regular graph** is a type of graph in which each vertex has the same number of edges, or connections, to other vertices. The degree of each vertex in a regular graph is constant. There are two main types of regular graphs: 1. **k-regular graph**: A graph is called k-regular (or simply regular) if every vertex has degree k.

Strongly regular graphs are a special class of graphs characterized by their regularity and specific connection properties between vertices. A graph \( G \) is called strongly regular with parameters \( (n, k, \lambda, \mu) \) if it satisfies the following conditions: 1. **Regularity**: The graph has \( n \) vertices, and each vertex has degree \( k \) (i.e., it is \( k \)-regular).

The Iofinova–Ivanov graph is a type of vertex-transitive graph that is defined using a specific set of rules based on combinatorial properties. The 110-vertex version of this graph specifically contains 110 vertices and has edges defined through particular mathematical relationships.

The Andrásfai graph is a specific type of graph in graph theory, distinguished for its properties related to being a strongly regular graph. It is named after the Hungarian mathematician János Andrásfai, who studied these structures. ### Properties of the Andrásfai Graph: 1. **Vertices and Edges**: The Andrásfai graph contains 18 vertices and 27 edges.

An antiprism graph is a geometric representation of a three-dimensional shape known as an antiprism. An antiprism is a polyhedron characterized by having two parallel polygonal bases connected by a band of triangular faces. The most common type of antiprism is the regular antiprism, where the bases are congruent regular polygons and the triangular faces are also isosceles triangles. In graph theory, the antiprism graph can be represented as a bipartite graph.

An Archimedean graph is a type of mathematical structure that is related to the concept of Archimedean solids in geometry. Specifically, an Archimedean graph is a vertex-transitive graph that can be represented as a Cayley graph of a group related to an Archimedean solid. In the context of geometry and polyhedra, Archimedean solids are convex polyhedra that are made up of two or more types of regular polygons.

The Balaban 10-cage is a specific type of polyhedron that is notable in the study of geometric structures and graph theory. It is a 3-dimensional shape that can be categorized as a type of cage, which is a regular polyhedron that has certain properties related to its vertices, edges, and faces.

The Balaban 11-cage is a type of graph that is part of a family of structures known as cage graphs. Cages are defined as regular graphs that have the smallest possible number of edges for a given number of vertices and girth (the length of the shortest cycle in the graph). Specifically, the Balaban 11-cage is an (11, 3)-cage, meaning it has 11 vertices and a girth of 3.

The Barnette–Bosák–Lederberg graph is an interesting example of a specific type of graph in the field of graph theory. It is notably a 3-connected cubic graph, meaning that it is a graph where each vertex has degree 3 (cubic) and it cannot be disconnected by removing just two vertices (3-connected). This graph is particularly recognized for having properties that make it an important object of study in relation to Hamiltonian paths and cycles.

The Bidiakis cube, also known as the Bidiakis knot, is a mathematical construct and a type of geometric puzzle. It is a variation of a cube that is often used in the study of topology and knot theory. The Bidiakis cube can also refer to a specific configuration of a geometric object where the cube exhibits certain twisting or knot-like properties, making it a subject of interest in mathematical visualization and education.

The Biggs–Smith graph is a specific type of graph in graph theory. It is defined as a 2-regular graph with 12 vertices and 12 edges. A 2-regular graph means that each vertex has a degree of 2, which implies that the graph consists of disjoint cycles.

Blanuša snarks are a specific type of snark, which is a type of non-trivial, 3-regular (each vertex has degree 3), edge-colored graph that lacks any homomorphic mapping to a 3-colorable graph, thus making it non-colorable with three colors. These graphs are named after the Croatian mathematician Josip Blanuša, who discovered them.

The Brinkmann graph is a specific type of graph in graph theory known for its unique properties. It is characterized as a 3-regular (cubic) graph, meaning that each vertex has exactly three edges connected to it.

In graph theory, a **cage** is a special type of graph that is defined by certain properties related to its vertices and edges. Specifically, a cage is a regular graph (a graph where each vertex has the same degree) with the fewest number of edges for a given degree and a specified girth (the length of the shortest cycle in the graph).

A Cameron graph is a specific type of graph that arises in combinatorics, particularly in the context of certain problems in graph theory and design theory. However, the term "Cameron graph" is not widely recognized in mathematical literature as a standard concept. It is possible that it refers to a specific graph or class of graphs studied by mathematicians like R. C. Cameron, who has made contributions to combinatorial designs and related areas.

Chang graphs, also known as Chang's graph or Chang's construction, are specific types of graphs in the field of combinatorial mathematics, particularly in graph theory. They are named after the mathematician Cheng-Chung Chang who introduced them in the context of studying properties of graphs and their applications in various areas of mathematics and computer science.

The Chvátal graph is a specific type of graph in the field of graph theory. It is a simple, undirected graph that consists of 12 vertices and 30 edges. The Chvátal graph is notable for several properties: 1. **Hamiltonian**: The Chvátal graph has a Hamiltonian cycle, meaning there exists a cycle that visits every vertex exactly once and returns to the starting vertex.

A **circulant graph** is a specific type of graph that generalizes the concept of cyclic graphs. It is defined using a description based on its vertex set and a set of connections (edges) determined by a set of step sizes.

Circular coloring is a concept in graph theory, specifically in the area of graph coloring. Unlike traditional graph coloring, where vertices of a graph are colored such that no two adjacent vertices share the same color, circular coloring allows for a more flexible coloring scheme: instead of using discrete colors, it uses a continuous spectrum of colors represented on a circle. In circular coloring, each vertex is assigned a position on the circumference of a circle, which corresponds to a color on a continuous scale.

The Clebsch graph is a specific type of graph in graph theory, notable for its unique mathematical properties. It has 16 vertices and 40 edges. The Clebsch graph can be described as a regular graph, meaning that each vertex has the same degree; specifically, each vertex in the Clebsch graph has a degree of 5.

A **complete graph** is a type of graph in which every pair of distinct vertices is connected by a unique edge. Complete graphs are denoted by the symbol \( K_n \), where \( n \) represents the number of vertices in the graph.

The Coxeter graph is an important concept in the fields of algebra, geometry, and graph theory. Specifically, it is a particular type of graph that represents the symmetric group and the properties of certain mathematical structures, particularly in relation to Coxeter groups. Here are some key features of the Coxeter graph: 1. **Definition and Structure**: The Coxeter graph is a finite undirected graph with 12 vertices and 18 edges.

A crown graph is a specific type of graph in graph theory. It is denoted as \( C_n \) and is defined as the graph that consists of two cycles \( C_n \) and \( C_{n+1} \) that are connected in a certain way. More formally, a crown graph can be defined as follows: 1. **Vertices**: The crown graph has \( 2n \) vertices, which can be represented as two disjoint cycles.

Cube-connected cycles (CCC) is a network topology used in parallel computing and interconnecting processing elements. It is a hybrid structure that combines features of both the hypercube network and cyclical connections. The primary purpose of CCC is to facilitate efficient communication between multiple processors in a system, making it suitable for parallel processing and distributed computing environments.

A **cubic graph**, also known as a **3-regular graph**, is a type of graph in which every vertex has a degree of exactly three. This means that each vertex is connected to exactly three edges. Cubic graphs are an important class of graphs in graph theory and have various applications in computer science, network design, and combinatorial optimization. ### Properties of Cubic Graphs: 1. **Degree**: Each vertex has a degree of 3.

A cycle graph, often denoted as \( C_n \), is a type of graph in which a set of vertices are connected in a closed loop. Specifically, in a cycle graph with \( n \) vertices, each vertex is connected to exactly two other vertices, creating a single cycle.

The term "Dejter graph" might not be widely recognized in the mathematical or graph theory communities. It is possible that it is a misspelling or a less common term. If you are referring to a well-known concept or a specific type of graph, please provide additional context or check the spelling. Some possible related terms could include "De Bruijn graph," "Dijkstra's graph," or "Directed graph," among others.

The Desargues graph is a finite, undirected graph named after the French mathematician Gérard Desargues. It is a special type of combinatorial structure that has connections to projective geometry and graph theory. The Desargues graph can be defined as follows: 1. **Vertices**: The graph has 20 vertices, which can be represented as points in a projective plane of order 2. 2. **Edges**: The graph has 30 edges.

The term "diamond cubic" refers to a specific crystal structure that is characteristic of diamond and several other materials, including silicon and germanium. In this structure, each carbon atom in diamond is covalently bonded to four other carbon atoms, resulting in a three-dimensional network.

A dipole graph is a specific type of graph used in physics and mathematics to represent a system featuring two opposing charges or poles, typically illustrated in the context of electric or magnetic fields. In the context of electrostatics, for example, a dipole consists of two point charges of equal magnitude and opposite sign separated by a distance.

Double-star snark refers to a specific kind of humor or sarcasm commonly found in the realm of online conversations, particularly in fan communities or discussions about various forms of media such as literature, movies, or video games. The term "snark" itself typically conveys a cutting, witty, or clever form of critique or commentary that can be both humorous and insightful.

A Dyck graph is a type of graph that represents the relationships between different valid sequences of balanced parentheses or paths in a lattice. The concept is often tied to combinatorial structures and is particularly connected to Dyck words, which are sequences of symbols that maintain a balance (for every opening symbol, there is a corresponding closing symbol).

The Dürer graph is a specific type of graph in the field of graph theory, named after the German painter and printmaker Albrecht Dürer. It is a highly symmetrical graph that has 12 vertices and 24 edges. The graph can be represented as a 3-dimensional object, which resembles a cube, and it is known for its interesting geometric properties.

The Ellingham-Horton graph is a thermodynamic reference tool used in metallurgy and materials science. It provides a visual representation of the standard free energy changes (ΔG) of various metal oxides as a function of temperature. Named after the researchers Sir Harold Ellingham and J. H. Horton, the graph is primarily used to analyze the stability of metal oxides and their tendency to reduce (or be reduced to their elemental form) at given temperatures.

The F26A graph is a specific type of graph used in the context of graph theory. It is commonly referenced as a particular standard graph that has a specific structure, often used in discussions of properties such as planarity, connectivity, and colorability. The F26A graph is often denoted within standard graph classifications and may have applications in various mathematical and computational contexts.

"Flower snark" typically refers to a playful or humorous type of sarcasm or witty commentary centered around flowers, gardening, or the aesthetics associated with them. It can manifest in various ways, such as funny social media posts, witty remarks about plant care, or tongue-in-cheek observations about floral design and gardening trends.

The folded cube graph is a type of mathematical graph that can be derived from the hypercube graph, particularly useful in the field of combinatorial design and graph theory. The concept is particularly involved in the analysis of topology, network design, and parallel processing. ### Definition: The \(n\)-dimensional folded cube graph, denoted \(FQ_n\), is constructed from the \(n\)-dimensional hypercube \(Q_n\).

A Folkman graph is a specific type of graph in graph theory named after the mathematician Julian Folkman. It is characterized by its properties related to edge connectivity and its structure. One important aspect of Folkman graphs is that they are used to investigate the relationship between graph properties such as colorings and connectivity. Specifically, Folkman graphs can be employed in studies related to hypergraphs and their extensions, especially in the context of coloring problems in combinatorial mathematics.

A Foster cage is a type of enclosure commonly used in biological research and veterinary settings to house animals. Named after biologist John Foster, these cages are designed to provide a controlled environment for animals, often for purposes such as observation, experimentation, or breeding. Foster cages are typically made from materials that allow for easy cleaning, visibility, and airflow.

The Foster graph is a specific type of graph in the field of graph theory. It is characterized as a bipartite graph with 12 vertices and 18 edges. The vertices can be divided into two disjoint sets, and every edge connects a vertex from one set to a vertex in the other set. The importance of the Foster graph arises from its role in various areas of graph theory, such as in the study of graph properties and structures, including colorability and chromatic polynomials.

The Franklin graph is a specific type of mathematical graph named after the American polymath Benjamin Franklin. It is notable for being a 12-vertex, 18-edge graph that can be geometrically embedded in three-dimensional space without any edges crossing. The Franklin graph is often used in the study of topology and graph theory due to its interesting properties. One of the notable features of the Franklin graph is its connectivity; it is 3-connected, meaning that removing any two vertices will not disconnect the graph.

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.

Klein graphs, or Klein four graphs, refer to a mathematical concept involving a specific type of graph related to group theory. The most referenced Klein graph is the **Klein four-group**, often denoted as \( V_4 \) or \( K_4 \). This is a group consisting of four elements that can be represented as the additive group of the vector space over the field with two elements.

A Kneser graph \( K(n, k) \) is a graph defined using the combinatorial structure of sets. Specifically, it is constructed from the set of all \( k \)-element subsets of an \( n \)-element set. The vertices of the Kneser graph correspond to these \( k \)-element subsets, and two vertices (i.e., subsets) are adjacent if and only if the corresponding subsets are disjoint.

The term "Laves graph" does not refer to a widely recognized concept in mathematics, graph theory, or any other standard academic discipline. However, it may be related to certain concepts in materials science, specifically Laves phases. Laves phases are types of intermetallic compounds that typically have a specific crystal structure and are significant in the study of alloys and solid materials.

A Livingstone graph is a type of mathematical graph used in the field of graph theory, specifically in relation to the study of networks and topological structures. It is named after the mathematician William Livingstone, though the term may not be widely recognized in all mathematical literature. Livingstone graphs are characterized by certain properties unique to their structure, often being studied for their applications in biology, chemistry, and network design.

The Ljubljana graph is a specialized graph in the field of graph theory. Specifically, it is a certain type of cubic (or 3-regular) graph, meaning that each vertex has exactly three edges connected to it. The Ljubljana graph is defined by a specific arrangement of vertices and edges, and it has some interesting properties, including being a distance-regular graph. It can be characterized by its vertex set and its connections, which lead to various applications in combinatorial designs and network theory.

The McGee graph is a specific type of graph in the field of graph theory. It is a 12-vertex, 18-edge undirected graph that can be constructed using certain properties of dual polyhedra. The McGee graph is notable for being a bipartite graph as well as a cubic graph, meaning that all its vertices have a degree of 3.

The McKay–Miller–Širáň graph is a notable bipartite graph that is specifically defined for its unique properties. It is a strongly regular graph, characterized as a (0, 1)-matrix representation. Key properties of this graph include: 1. **Vertex Count**: It has a total of 50 vertices. 2. **Regularity**: Each vertex connects to exactly 22 other vertices.

The McLaughlin graph is a particular type of graph in the field of graph theory. It is an undirected graph that has some interesting properties and is often studied in relation to cliques, colorings, and various other graph properties. Here are some key characteristics of the McLaughlin graph: 1. **Vertices and Edges**: The McLaughlin graph has 12 vertices and 30 edges.

The Meredith graph is a specific type of graph in the field of graph theory. It is defined as a bipartite graph and is notable because it is a regular graph with 12 vertices, where each vertex has a degree of 3. The graph consists of two sets of vertices, each containing 6 vertices, and it can be described by specific connections between these two sets.

A Meringer graph is a specific type of mathematical graph that is known for its unique properties related to vertex connectivity. The Meringer graphs are typically constructed using certain combinatorial techniques and can serve as examples in graph theory studies. One of the notable features of Meringer graphs is that they can be used to demonstrate various aspects of connectivity, cycles, and other graph properties.

A Moore graph is a special type of undirected graph that has particular properties related to its diameter, degree, and the number of vertices. Specifically, a Moore graph is defined as a regular graph of degree \( k \) with diameter \( d \) that has the maximum possible number of vertices for those parameters.

The Möbius ladder is a type of geometric structure that combines concepts from topology and graph theory. Specifically, it is a type of graph that can be visualized as a ladder with a twist, similar to the famous Möbius strip.

The Möbius–Kantor graph is a specific type of graph that arises in the context of projective geometry and has interesting combinatorial properties. It can be described as follows: 1. **Vertices**: The Möbius–Kantor graph has 12 vertices. These can be thought of as corresponding to the 12 lines of the projective plane over the field with two elements.

The Nauru graph is a specific type of graph in the field of graph theory. It is notably characterized as a **strongly regular graph**, which means it has a certain degree of regularity in its structure.

A **null graph** (also known as the **empty graph**) is a type of graph in graph theory that contains no vertices and therefore no edges. In other words, it is a graph that has no points or connections between them. Alternatively, when talking about a more general context in graphs that do involve vertices, a null graph can also refer to a graph that has vertices but no edges connecting any of them.

In graph theory, an **odd graph** often refers to a specific type of graph constructed from a complete graph by removing certain edges. One common interpretation of an odd graph is as follows: 1. **Odd Cycle Graph**: A cycle graph with an odd number of vertices (e.g. a triangle, pentagon, heptagon, etc.) is known as an odd cycle graph.

A Paley graph is a specific type of mathematical graph that is constructed from a finite field. It is named after the mathematician Arthur Paley. Paley graphs are particularly interesting in the fields of combinatorics and number theory, and they have applications in areas such as coding theory and the design of networks. ### Construction of Paley Graphs 1.

The Pappus graph is a specific type of cubic graph that has a number of interesting properties in the field of graph theory. It is named after the ancient Greek mathematician Pappus of Alexandria. Here are some key characteristics of the Pappus graph: - **Vertices and Edges**: The Pappus graph has 18 vertices and 27 edges. - **Cubic Graph**: It is a cubic graph, meaning that each vertex has a degree of 3.

A Perkel graph is a special type of graph used in the study of graph theory and combinatorial designs. It is defined based on a recursive structure. Specifically, a Perkel graph is constructed from an initial set of vertices and uses certain rules to add edges based on the properties of those vertices.

The Petersen graph is a well-known and important object in the field of graph theory. It is a specific undirected graph that has several interesting properties. Here are some key features of the Petersen graph: 1. **Vertices and Edges**: The Petersen graph consists of 10 vertices and 15 edges.

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 quartic graph is a graphical representation of a polynomial function of degree four.

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.

The Tutte 12-cage is a specific type of graph in the field of graph theory, named after the mathematician W.T. Tutte. It is notable for being a strongly regular graph with particular properties. ### Characteristics of the Tutte 12-Cage: 1. **Vertices and Edges**: It has 12 vertices and 30 edges.

The Tutte graph is a specific, well-known example of a cubic graph (3-regular graph) that is often studied in the field of graph theory. It has several interesting properties and characteristics: 1. **Vertices and Edges**: The Tutte graph has 46 vertices and 69 edges. It is one of the smallest cubic graphs that is not 3-colorable, meaning it cannot be colored with three colors without two adjacent vertices sharing the same color.

The Tutte–Coxeter graph is a well-known graph in the study of graph theory and combinatorics. It is a bipartite graph with some interesting properties and significance. Here are some key features of the Tutte–Coxeter graph: 1. **Vertices and Edges**: The Tutte–Coxeter graph consists of 12 vertices and 18 edges.

The Wagner graph is a specific type of undirected graph that is notable in the study of graph theory. It has 12 vertices and 30 edges, and it is characterized by being both cubic (each vertex has a degree of 3) and 3-regular. One of the most interesting properties of the Wagner graph is that it is a non-planar graph, meaning it cannot be drawn on a plane without edges crossing.

Watkins snark, also known as "watkins snark," typically refers to a specific type of mathematical problem or concept explored in various fields of combinatorics and graph theory. Unfortunately, there isn't a widely recognized definition for "Watkins snark"; it's possible that it could be a niche term or a recent development in a specialized area of mathematics.

In graph theory, a Wells graph is a specific type of graph that is defined based on the properties of certain combinatorial structures. Specifically, Wells graphs arise in the context of geometric representation of graphs and are related to the concept of unit distance graphs. A Wells graph is characterized by its degree of vertex connectivity and geometric properties, particularly in higher-dimensional spaces. It often finds applications in problems involving networking, combinatorial designs, and the study of geometric configurations.

A Wong graph is a specific type of directed graph that is used in graph theory, named after the mathematician David Wong who introduced it. The defining characteristic of a Wong graph is its ability to model certain kinds of dependency relations and interactions between nodes.

Adjacency algebra is a mathematical framework used primarily in the field of graph theory and network analysis. It focuses on the representation and manipulation of graphs using algebraic techniques. The core concept of adjacency algebra revolves around the adjacency matrix of a graph, which is a square matrix used to represent a finite graph.

An adjacency matrix is a square matrix used to represent a finite graph. It indicates whether pairs of vertices (or nodes) in the graph are adjacent (i.e., connected directly by an edge) or not. Here's how it works: 1. **Matrix Structure**: The size of the adjacency matrix is \( n \times n \), where \( n \) is the number of vertices in the graph. Each row and column of the matrix corresponds to a vertex in the graph.

Algebraic connectivity is a concept from graph theory that measures the connectivity of a graph in a specific way. It is defined as the smallest non-zero eigenvalue of the Laplacian matrix of a connected graph.

The Alon–Boppana bound is a result in the field of graph theory and spectral graph theory. It provides a lower bound on the largest eigenvalue (also known as the spectral radius) of a regular graph. More formally, let \( G \) be a \( d \)-regular graph on \( n \) vertices.

Babai's problem, named after mathematician László Babai, is a computational problem related to the field of group theory and complexity theory, particularly in the context of lattice problems. The problem specifically deals with the challenge of finding the closest lattice vector to a given point in high-dimensional space.

Brouwer's conjecture, proposed by the Dutch mathematician L.E.J. Brouwer in the early 20th century, is a statement in the field of topology, particularly concerning the nature of continuous functions and fixed points. Specifically, the conjecture asserts that every continuous function from a compact convex set to itself has at least one fixed point.

Centrality is a concept used in various fields, including mathematics, network theory, sociology, and data analysis, to measure the importance or influence of a node (such as a person, organization, or computer) within a network. The idea is that some nodes hold more power or are more significant than others based on their position and connections within the network.

The clustering coefficient is a measure used in network theory to quantify the degree to which nodes in a graph tend to cluster together. It provides a way to understand the local structure of a network. There are two main types of clustering coefficients: the local clustering coefficient and the global clustering coefficient.

The complex network zeta function is a mathematical tool used in the study of complex networks, which are structures characterized by interconnected nodes (or vertices) and edges (or links). This zeta function is often associated with certain properties of the network, such as its topology, dynamics, or spectral characteristics. ### Key Concepts 1. **Complex Networks**: These are graphs with complex structures, which can represent various real-world systems, such as social networks, transportation systems, biological networks, etc.

In graph theory, conductance is a measure that indicates how well a graph can conduct flow between its parts. It is typically used in the context of studying random walks or the mixing properties of a graph. Conductance helps understand how well connected different regions (or communities) of a graph are.

A **conference graph** is a specific type of graph studied in graph theory, related to combinatorial designs.

A conference matrix is a concept mainly used in combinatorics, specifically in the study of error-correcting codes, design theory, and graph theory. It is related to structured arrangements of points and lines, usually in the context of finite groups and their applications. More formally, a conference matrix is an \( n \times n \) matrix, where \( n \) is an even integer, that has specific properties: 1. The entries of the matrix are either 0 or 1.

In graph theory, a **cycle basis** of a graph is a minimal set of cycles such that any cycle in the graph can be expressed as a combination of these cycles. Specifically, for a connected graph, a cycle basis serves as a framework for the cycles of the graph. ### Key Points: 1. **Cycles**: A cycle in a graph is a path that starts and ends at the same vertex, with no other vertices repeated.

In the context of graph theory, a **cycle space** is a fundamental concept associated with the study of cycles in graphs. Specifically, it is a vector space formed by the cycles of a graph when considered over a field (typically the field of two elements, often denoted as GF(2)). Here’s a more detailed breakdown: 1. **Graph Basics**: A graph is defined as a collection of vertices (or nodes) connected by edges.

In the context of graph theory, the degree matrix is a square diagonal matrix that is used to represent the degrees of the vertices in a graph. Specifically, for a simple undirected graph \( G \) with \( n \) vertices, the degree matrix \( D \) is defined as follows: 1. The matrix \( D \) is of size \( n \times n \). 2. The diagonal entries of \( D \) are the degrees of the corresponding vertices in the graph.

A distance-regular graph is a specific type of graph that has a high degree of regularity in the distances between pairs of vertices. Formally, a graph \( G \) is said to be distance-regular if it satisfies the following conditions: 1. **Regularity**: The graph is \( k \)-regular, meaning each vertex has exactly \( k \) neighbors.

A distance-transitive graph is a type of graph that exhibits a high degree of symmetry with respect to the distances between vertices.

In graph theory, a dual graph is a construction that relates to a planar graph. To understand dual graphs, it's important to start with the concept of a planar graph itself. A planar graph is a graph that can be drawn on a plane without any edges crossing. ### Key Concepts of Dual Graphs 1. **Vertices of the Dual Graph**: For every face (region) in the original planar graph, there is a corresponding vertex in the dual graph.

An **edge-transitive graph** is a type of graph that has a high degree of symmetry. Specifically, a graph is called edge-transitive if, for any two edges in the graph, there exists an automorphism (a graph isomorphism from the graph to itself) that maps one edge to the other. This means that all edges of the graph are essentially indistinguishable in terms of the structure of the graph.

In the context of graph theory and computational mathematics, edge and vertex spaces can refer to the associated vector spaces constructed from the edges and vertices of a graph. These concepts are often utilized in the study of networks, combinatorial structures, and various applications in computer science and mathematics.

Edmonds matrix is a mathematical concept used in the context of graph theory and combinatorial optimization, particularly in relation to the Edmonds-Karp algorithm for finding maximum flows in flow networks. However, some confusion arises because the term might also relate to different objects depending on the context. 1. **In Graph Theory**: The Edmonds matrix is sometimes referred to in discussions of cut matrices or adjacency matrices related to specific types of graphs.

Elementary Number Theory, Group Theory, and Ramanujan Graphs are three distinct yet important topics in mathematics, particularly in the fields of number theory, algebra, and graph theory. Here's a brief overview of each: ### Elementary Number Theory Elementary number theory is a branch of mathematics that deals with the properties and relationships of numbers, particularly integers. It does not involve advanced mathematical tools such as calculus or abstract algebra.

The Expander Mixing Lemma is a result from the field of graph theory, particularly in the study of expander graphs. Expander graphs are sparse graphs that have strong connectivity properties, which makes them useful in various applications, including computer science, combinatorics, and information theory. The Expander Mixing Lemma provides a quantitative measure of how well an expander graph mixes the vertices when performing random walks on the graph.

Frucht's theorem is a result in graph theory that states that for any finite group \( G \), there exists a finite undirected graph (called a "Frucht graph") that is a Cayley graph of \( G \) and is also vertex-transitive (meaning that for any two vertices in the graph, there is some automorphism of the graph that maps one vertex to the other).

The Graham–Pollak theorem is a result in graph theory that pertains to the relationships between the edges of a complete graph and the configurations of points in Euclidean space. Specifically, it states that for a complete graph on \( n \) vertices, the number of edges that can be embedded in \( \mathbb{R}^d \) (real d-dimensional space) without any three edges crossing is limited.

Graph automorphism is a concept in graph theory that refers to a symmetry of a graph that preserves its structure. More specifically, an automorphism of a graph is a bijection (one-to-one and onto mapping) from the set of vertices of the graph to itself that preserves the adjacency relationship between vertices.

Graph energy is a concept from spectral graph theory, which is a field of mathematics that studies graphs through the properties of matrices associated with them. Specifically, graph energy is related to the eigenvalues of a graph's adjacency matrix.

The Hafnian is a mathematical function related to the theory of matrices and combinatorial structures. Specifically, it can be viewed as a generalization of the permanent of a matrix. For a given \( n \times n \) matrix \( A = [a_{ij}] \), the hafnian is defined only for matrices of even order, \( n = 2k \).

A **half-transitive graph** is a type of graph that is related to the concept of transitive graphs in the field of graph theory. To understand half-transitive graphs, it's helpful to first clarify what a transitive graph is.

Hierarchical closeness typically refers to a concept in social network analysis and organizational theory that measures how closely related individuals or entities are within a hierarchical structure based on their positions. It can be used to assess the proximity of nodes (which could represent people, departments, or other entities) within social or organizational hierarchies.

The Ihara zeta function is a mathematical object that arises in the study of finite graphs, particularly in the context of algebraic topology and number theory. It was introduced by Yoshio Ihara in the 1960s.

An incidence matrix is a mathematical representation used primarily in graph theory and related fields to represent the relationship between two classes of objects. In the context of graph theory, an incidence matrix is used to describe the relationship between vertices (nodes) and edges in a graph.

An **integral graph** is a type of graph in which all of its eigenvalues are integers. The eigenvalues of a graph are derived from its adjacency matrix, which represents the connections between the vertices in the graph.

The Jordan–Pólya number is a concept from the field of mathematics, particularly in number theory and combinatorial mathematics. It is defined as a non-negative integer that can be expressed as the sum of distinct positive integers raised to a power that increases with each integer.

Katz centrality is a measure of the relative influence of a node within a network. It extends the concept of degree centrality by considering not just the immediate connections (i.e., the direct neighbors of a node) but also the broader network, taking into account the influence of nodes that are connected to a node's neighbors. The fundamental idea behind Katz centrality is that a node is considered important not only because it has many direct connections but also because its connections lead to other connected nodes.

Kirchhoff's theorem can refer to several concepts in different fields of physics and mathematics, but it is most commonly associated with Kirchhoff's laws in electrical circuits and also with a theorem in graph theory. 1. **Kirchhoff's Laws in Electrical Engineering**: - **Kirchhoff’s Current Law (KCL)**: This law states that the total current entering a junction in an electrical circuit equals the total current leaving the junction.

The Laplacian matrix is a representation of a graph that encodes information about its structure and connectivity. It is particularly useful in various applications such as spectral graph theory, machine learning, image processing, and more.

The Lovász conjecture is a well-known conjecture in combinatorial discrete mathematics, specifically in the field of graph theory. Proposed by László Lovász in 1970, the conjecture pertains to the structure of edge-coloring in a certain class of graphs known as Kneser graphs. To explain the conjecture, we first need to define Kneser graphs.

Mac Lane's planarity criterion, also known as the "Mac Lane's formation", is a combinatorial condition used to determine whether a graph can be embedded in the plane without any edges crossing. Specifically, the criterion states that a graph is planar if and only if it does not contain a specific type of subgraph as a "minor.

The matching polynomial is a well-defined polynomial associated with a graph that encapsulates information about its matchings—sets of edges without shared vertices.

The minimum rank of a graph is a concept from algebraic graph theory that is associated with the graph's adjacency matrix or Laplacian matrix. Specifically, it refers to the smallest rank among all real symmetric matrices corresponding to the graph.

Modularity, in the context of networks, refers to the degree to which a network can be divided into smaller, disconnected sub-networks or communities. It is often used in network analysis to identify and measure the strength of division of a network into modules, which are groups of nodes that are more densely connected to each other than to nodes in other groups. ### Key Points about Modularity: 1. **Community Structure**: Modularity helps in detecting community structure within networks.

The Parry–Sullivan invariant is a concept in the field of dynamical systems and statistical mechanics, particularly related to the study of interval exchanges and translations. It is associated with the study of the dynamics of certain classes of transformations, particularly those that exhibit specific structural and statistical properties. The invariant itself is often connected to topological and measure-theoretic characteristics of systems that exhibit a certain type of symmetry or recurrence.

A Ramanujan graph is a type of expander graph named after the Indian mathematician Srinivasa Ramanujan, whose work in number theory inspired this concept. Ramanujan graphs are particularly characterized by their exceptional expansion properties and have applications in various areas of mathematics and computer science, including combinatorics, number theory, and network theory.

In graph theory, the term "rank" can have a couple of different meanings, depending on the context in which it is used. 1. **Rank of a Graph**: The rank of a graph can refer to the maximum number of edges that can be included in a spanning tree. In this context, it is often considered in relation to the concept of the graph's connectivity and the number of vertices (V) and edges (E).

A Seidel adjacency matrix is a type of matrix used in graph theory, particularly for the representation of certain types of graphs known as Seidel graphs. It is derived from the standard adjacency matrix of a graph but has a distinctive form.

A semi-symmetric graph is a type of graph that exhibits certain symmetrical properties but does not necessarily exhibit full symmetry. More formally, a semi-symmetric graph can be defined through its vertex and edge structure in relation to their automorphisms and symmetrical actions. In the context of graph theory, the properties that characterize a semi-symmetric graph can vary somewhat depending on the specific definition being used. Generally, however, a semi-symmetric graph maintains some degree of regularity and uniformity in its structure.

Sims' conjecture is a hypothesis in the field of algebraic topology and combinatorial group theory, specifically relating to the properties of certain types of groups. Named after mathematician Charles Sims, the conjecture primarily deals with the structure of finite groups and representation theory. While specific details or formulations may vary, Sims' conjecture is generally focused on establishing a relationship between the orders of groups and their representations or modules.

Spectral clustering is a technique used in machine learning and data analysis for grouping data points into clusters based on the properties of the dataset. It leverages the eigenvalues and eigenvectors of matrices derived from the data, particularly the similarity matrix, to identify clusters. Here’s an overview of the key steps and concepts involved in spectral clustering: 1. **Similarity Graph**: First, a similarity graph is constructed from the data points.

Spectral graph theory is a branch of mathematics that studies the properties of graphs through the eigenvalues and eigenvectors of matrices associated with them. These matrices include the adjacency matrix, the degree matrix, and the Laplacian matrix, among others. Spectral graph theory connects combinatorial properties of graphs with linear algebra and provides powerful tools for analyzing graphs in various contexts.

A strongly regular graph is a specific type of graph characterized by a regular structure that satisfies certain conditions regarding its vertices and edges. Formally, a strongly regular graph \( G \) is defined by three parameters \( (n, k, \lambda, \mu) \) where: - \( n \) is the total number of vertices in the graph.

A symmetric graph is a type of graph that exhibits a certain level of symmetry in its structure. More formally, a graph \( G \) is considered symmetric if, for any two vertices \( u \) and \( v \) in \( G \), there is an automorphism of the graph that maps \( u \) to \( v \).

The Tutte matrix is a mathematical construct used in the study of graph theory, particularly in the context of understanding the properties of bipartite graphs and the presence of perfect matchings. It is named after the mathematician W. T. Tutte.

A "two-graph" typically refers to a specific type of graph in the field of graph theory, but it might not be a widely standardized term. In general, graph theory involves studying structures made up of vertices (or nodes) connected by edges.

A **vertex-transitive graph** is a type of graph in which, for any two vertices, there is some automorphism of the graph that maps one vertex to the other. In simpler terms, this means that the graph looks the same from the perspective of any vertex; all vertices have a similar structural role within the graph. ### Key Properties: 1. **Automorphism:** An automorphism is a bijection (one-to-one correspondence) from the graph to itself that preserves the edges.

A **walk-regular graph** is a type of graph that has a uniform structure relative to walks of certain lengths. Specifically, a graph is called \( k \)-walk-regular if the number of walks of length \( k \) from any vertex \( u \) to any other vertex \( v \) depends only on the distance between \( u \) and \( v \), rather than on the specific choice of \( u \) and \( v \).