Graphs

Graphs are mathematical structures used to model pairwise relationships between objects. They consist of vertices (or nodes) and edges (connections between the vertices). Graphs can be used to represent various systems in numerous fields, including computer science, social science, biology, and transportation. ### Key Terminology: 1. **Vertices (or Nodes)**: The fundamental units or points of the graph. They can represent entities such as people, cities, or any discrete items.

Application-specific graphs are specialized data structures that are tailored to represent relationships and interactions specific to a particular domain, application, or problem scenario. Unlike general-purpose graphs, which can represent a wide range of relationships in an abstract manner, application-specific graphs are designed to optimize for certain characteristics, constraints, or patterns inherent in the particular application they serve.

Graph data structures are a mathematical representation used to capture relationships between pairs of objects. A graph consists of two fundamental components: 1. **Vertices (or Nodes)**: These are the individual entities or points in the graph. Each vertex represents an object in the data set. 2. **Edges**: These are the connections or relationships between the vertices. An edge can connect two vertices and may represent various types of associations, such as friendship in social networks, pathways in maps, or relationships in databases.

In computer science, a **tree** is a widely-used data structure that simulates a hierarchical organization of data. It consists of nodes connected by edges and has the following key characteristics: 1. **Roots and Nodes**: - The topmost node is called the *root*. - Each node can have zero or more child nodes (subsequent nodes branching from it). - A node that does not have any children is termed a *leaf* node.

A Bratteli diagram is a graphical representation used in the study of certain types of mathematical structures, particularly in the fields of operator algebras and dynamical systems. It is named after the mathematician Ole Bratteli, who introduced this concept. ### Key Features of Bratteli Diagrams: 1. **Structure**: A Bratteli diagram is a directed graph comprised of vertices and edges.

The Bratteli–Vershik diagram is a combinatorial and graphical representation used primarily in the study of dynamical systems, particularly in the context of partitioning and representing the structure of infinite-dimensional objects, such as representing the flow of certain dynamical systems or the actions of groups on spaces.

A citation graph is a directed graph that represents the relationship between academic papers, articles, patents, or other scholarly works based on citations. In a citation graph: - **Nodes**: Each node corresponds to a publication or scholarly work. - **Edges**: A directed edge from node A to node B indicates that publication A cites publication B. This means that A references or relies on B in its content.

As of my last knowledge update in October 2021, there was no well-known entity specifically referred to as the "Co-stardom network." It's possible that it's a new or niche initiative, organization, or platform that has emerged since then. If the term is related to a specific context, like entertainment, social media, or a particular community, additional details could help clarify its meaning.

The Code Property Graph (CPG) is a data structure that combines elements from abstract syntax trees (AST), control flow graphs (CFG), and call graphs to provide a comprehensive representation of source code. It is primarily used in the field of static analysis and program analysis to facilitate various tasks such as vulnerability detection, code quality assessment, and automated program verification. Key features of the Code Property Graph include: 1. **Unified Representation**: CPG integrates different representations of code into a single graph.

A collaboration graph is a type of visual representation that illustrates the relationships and interactions between individuals or entities involved in a collaborative effort or network. This can apply to various contexts, such as: 1. **Social Networks**: In social media or professional networking, a collaboration graph might show how different users or groups connect and interact with each other, highlighting relationships, common interactions, and the flow of information or resources.

A conceptual graph is a form of knowledge representation that is used to describe and visualize concepts and their relationships in a structured way. Developed by Roger Schank and later refined by others, conceptual graphs are particularly useful in artificial intelligence, natural language processing, and cognitive science. ### Components of Conceptual Graphs 1. **Nodes**: Represent concepts, entities, or objects. Each node can describe a specific concept or a category in the graph.

A configuration graph is a type of graph used to represent the states and transitions of a system, particularly in the context of distributed systems, robotics, or combinatorial problems. In general, configuration graphs help visualize how different configurations (or states) of a system can transition from one to another based on certain rules or actions.

A **constraint graph** is a graphical representation of a set of variables and the constraints that exist between them. It is particularly useful in fields like artificial intelligence, operations research, and computer science, especially for problems involving constraint satisfaction. ### Components of a Constraint Graph: 1. **Nodes (Vertices)**: Each node represents a variable in the problem. For example, if you are solving a scheduling problem, each variable could represent a specific task that needs to be scheduled.

A constraint graph is a graphical representation used in various fields such as computer science, mathematics, and operational research, particularly in the context of constraint satisfaction problems (CSPs) and layout problems. In the realm of layout design—such as for electronic circuits, user interfaces, or other spatial arrangements—a constraint graph helps illustrate the relationships and restrictions between different elements or components.

A control-flow graph (CFG) is a representation used in computer science and software engineering to model the flow of control within a program or a function. It is particularly useful in the fields of compiler design, static analysis, and program optimization. ### Key Characteristics of a Control-flow Graph: 1. **Nodes**: Each node in a CFG represents a basic block of code, which is a straight-line sequence of instructions with no control transfers (like jumps or branches) except at the start and end.

A disjunctive graph is a concept often encountered in the fields of graph theory and computer science, particularly in relation to representation and analysis of logical expressions, automata, and certain types of optimization problems. However, the term "disjunctive graph" is not universally defined, and its meaning can vary based on the context.

"Eodermdrome" does not appear to be a recognized term in the medical community or in scientific literature as of my last update. It may be a misspelling or a combination of terms related to dermatology, such as "derm" (skin) and "drome" (which often refers to a running or a syndrome).

Evolutionary graph theory is a subfield of mathematics that applies concepts from graph theory to understand the dynamics of evolutionary processes, particularly how populations evolve over time within structured environments. It combines elements of population genetics, evolutionary biology, and network theory. In evolutionary graph theory, individuals in a population are represented as vertices (or nodes) in a graph, and the edges (connections between nodes) represent the interactions or relationships between these individuals.

A factor graph is a type of bipartite graph used in statistics, probability, and machine learning to represent the factorization of a probability distribution. It provides a visual and structural way to denote how variables and factors (functions that define relationships between variables) are interconnected. ### Key Components: 1. **Variables**: These are typically represented as nodes on one side of the graph. Each variable can be a random variable in a probabilistic model.

A Graph-Structured Stack is a data structure that extends the traditional stack concept by organizing data elements in a graph format rather than a linear sequence. In a traditional stack, elements are added and removed in a Last-In-First-Out (LIFO) manner, where each element only has a single predecessor and successor. In contrast, a graph-structured stack allows for more complex relationships between elements.

The Ingredient-Flavor Network is a concept that explores the relationships between various food ingredients and their associated flavors. It is often represented as a network where ingredients serve as nodes and their flavor characteristics or pairings are represented as edges connecting these nodes. This network can help chefs, food scientists, and food enthusiasts to understand which ingredients complement each other based on shared flavor compounds or culinary traditions.

A "Loss Network" generally refers to a type of network in telecommunications and network theory where packet loss occurs, often due to congestion or other adverse conditions. This can be in the context of data networks, where data packets may be dropped, leading to a loss of information. In such networks, performance analysis is crucial because packet loss can significantly affect the quality of service (QoS) and overall network reliability.

A **prime graph** is a concept in the field of algebra, particularly in the study of group theory and graph theory. It is associated with the study of group actions, and it specifically relates to the representation of groups as graphs. The **prime graph** of a finite group \( G \) is constructed by representing the prime divisors of the order of the group as vertices in a graph.

A process graph is a visual representation of the steps and activities involved in a specific process. It helps to illustrate how different tasks interrelate, their sequence, and the flow of information or materials throughout the process. Process graphs are commonly used in various fields, including business process management, software development, systems engineering, and project management. Key components of a process graph typically include: 1. **Nodes**: Represent specific tasks, activities, or decision points in the process.

A Program Dependence Graph (PDG) is a graphical representation of the dependencies within a program, specifically focusing on the relationships between different computations and data in the program. PDGs are useful for various analyses and optimizations in compiler design and software engineering. ### Key Components of a PDG: 1. **Nodes:** - **Statements or Instructions:** Each node in the graph represents a basic operation or statement in the program.

A Reeb graph is a topological construct used in the study of continuous functions on manifolds. It is named after the mathematician George Reeb, who introduced it in the context of topology and differential geometry. Essentially, a Reeb graph captures the way that a continuous function "groups" points based on the level sets of the function.

A Tanner graph is a type of bipartite graph that is used to represent error-correcting codes, particularly low-density parity-check (LDPC) codes. Named after Michael Tanner, who introduced this representation in the 1980s, Tanner graphs provide a visual and mathematical way to describe the relationships between code symbols (variables) and parity-check constraints (checks) in coding theory.

A trellis in graph theory is a specific type of graph that is often used to represent the structure of a network or a mathematical object, particularly in the context of coding theory, data transmission, and signal processing.

Webgraph is a term that can refer to different concepts depending on the context, but generally, it is primarily associated with the representation and analysis of the link structure of the web. Here are some common interpretations of Webgraph: 1. **Web Structure Mining**: In the context of web structure mining, a Webgraph is a directed graph that represents the relationships between web pages. Each node in the graph corresponds to a web page, and each directed edge represents a hyperlink from one page to another.

In graph theory, "graph families" refer to groups or classes of graphs that share certain properties or characteristics. These families can be defined based on various criteria, including structural properties, combinatorial features, or applications. Understanding graph families helps in categorizing and analyzing graphs, allowing for more efficient algorithms and insights into their behavior. Here are some common types of graph families: 1. **Planar Graphs**: Graphs that can be drawn on a plane without any edges crossing.

The concept of intersection classes in graph theory refers to a way of classifying graphs based on their intersections with certain predefined properties or structural constraints. Typically, an intersection class is formed by taking the intersection of a set of graphs with a specific property or defining characteristic.

Parametric families of graphs refer to a collection of graphs defined by one or more parameters that can take on different values. These parameters can dictate various properties of the graphs, such as their structure, size, or constraints. Parametric families are useful in combinatorics, graph theory, and algorithm design because they allow researchers and practitioners to analyze a broad class of graphs simultaneously and to derive general results or algorithms that apply to all graphs within the family.

In graph theory, a **tree** is a type of graph that has specific properties. Here are the key characteristics that define a tree: 1. **Acyclic**: A tree is acyclic, meaning it does not contain any cycles. In other words, there is no way to start at one vertex, travel along the edges, and return to the same vertex without retracing steps.

An asymmetric graph, often referred to in the context of graph theory, typically means that the graph lacks symmetry in its structure. More formally, a graph is considered asymmetric if it does not have non-trivial automorphisms, which are mappings from the graph to itself that preserves the structure (i.e., the vertices and edges).

A bipartite graph is a specific type of graph in graph theory that can be divided into two distinct sets of vertices such that no two vertices within the same set are adjacent. In other words, the edges of a bipartite graph only connect vertices from one set to vertices from the other set.

A **bivariegated graph** is a specific type of graph in which the vertex set can be divided into two distinct sets such that no two vertices within the same set are adjacent. This means that every edge connects a vertex from one set to a vertex from the other set. In essence, a bivariegated graph is a bipartite graph.

A block graph is a type of graph that is particularly used in computer science and graph theory. It is a representation of a graph that groups vertices into blocks, where a block is a maximal connected subgraph that cannot be separated into smaller connected components by the removal of a single vertex. In simpler terms, blocks represent parts of the graph that are tightly connected and removing any one vertex from a block won't disconnect the block itself.

A bound graph is a concept primarily used in the context of graph theory, though the term can be related to various domains such as computer science, mathematics, and other related fields. However, "bound graph" may not refer to a widely recognized term with a single definition across various disciplines.

A **cactus graph** is a special type of graph in graph theory with a specific structural property. A cactus graph is defined as a connected graph in which any two cycles have at most one vertex in common. In simpler terms, while a cactus can have multiple cycles, these cycles cannot intersect in more than one vertex, meaning that their intersections (if any) do not create complex overlapping structures.

A **chordal bipartite graph** is a specific type of graph that has properties of both chordal graphs and bipartite graphs. 1. **Bipartite Graph:** A graph is called bipartite if its vertex set can be divided into two disjoint sets \( U \) and \( V \) such that no two vertices within the same set are adjacent.

A **chordal graph**, also known as a **cographic graph**, is a type of graph in which every cycle of four or more vertices has a chord. A **chord** is an edge that connects two non-adjacent vertices in a cycle.

A **cluster graph** is a type of graph in graph theory that consists of several complete subgraphs, known as clusters, that are connected by edges in a structured way. More specifically, it can be defined as follows: - **Clusters**: Each cluster is a complete graph where every pair of vertices within that cluster is connected by an edge. If a cluster has \(k\) vertices, it contains \( \frac{k(k-1)}{2} \) edges.

A comparability graph is a type of graph that arises in the field of graph theory, specifically in the study of ordered sets (partially ordered sets or posets). In a comparability graph, the vertices represent elements of a partially ordered set, and there is an edge between two vertices if and only if the corresponding elements are comparable in the poset. This means one element is either less than or greater than the other according to the ordering.

A **convex bipartite graph** is a specific type of graph that belongs to the category of bipartite graphs, which are graphs where the vertex set can be divided into two disjoint subsets such that every edge connects a vertex in one subset to a vertex in the other. In a bipartite graph, there are no edges between vertices within the same subset. The term **convex** typically relates to a property concerning the induced subgraphs of the bipartite graph.

A **dense graph** is a type of graph in which the number of edges is close to the maximal number of edges that can exist between the vertices. More formally, a graph is considered dense if the ratio of the number of edges \( E \) to the number of vertices \( V \) squared, \( \frac{E}{V^2} \), is relatively large.

A distance-hereditary graph is a type of graph in which the distances between pairs of vertices are preserved in all connected induced subgraphs.

A **dually chordal graph** is a type of graph that has specific structural properties related to both its vertices and cycles. The term "dually chordal" arises in the context of vertex or edge properties. 1. **Chordal Graph**: - A graph is called **chordal** if every cycle of length four or more has a chord. A chord is an edge that is not part of the cycle but connects two vertices of the cycle.

An **even-hole-free graph** is a type of graph in which there are no induced subgraphs that form a cycle of even length greater than 2, also known as an "even hole." In simpler terms, if a graph is even-hole-free, it does not contain a cycle that is both even (has an even number of edges) and cannot be extended by adding more edges or vertices without creating adjacent edges (i.e., it is an induced subgraph).

An **expander graph** is a type of sparse graph that has strong connectivity properties. More formally, it is a family of graphs that exhibit high expansion, meaning that they have a well-defined, large number of edges relative to the number of vertices.

In the context of mathematical logic and set theory, particularly in the area of model theory and set-theoretic topology, a **forcing graph** is not a standard term. However, it may refer to concepts related to forcing conditions in the context of set theory. **Forcing** is a technique introduced by Paul Cohen in the 1960s.

A **geodetic graph** is a type of graph in the field of graph theory, characterized by the property that any two distinct vertices in the graph are connected by a unique shortest path. In other words, for every pair of vertices in a geodetic graph, there exists exactly one geodesic (the shortest path) between them.

A Halin graph is a type of graph that is formed from a connected, planar graph, specifically by taking the dual of a polyhedron and then removing its outer face. It can also be constructed by taking a tree (specifically, a connected graph without cycles), doubling its edges, and connecting the resulting vertices to form a polyhedral structure. Halin graphs are named after Rudolf Halin, who contributed significantly to their study.

A Hanan grid is a specific type of geometric structure used in combinatorial optimization, particularly in the context of network design and facility location problems. Named after its creator, M. Hanan, it consists of a grid created from a given set of points (usually in a Euclidean space) by placing vertical and horizontal lines between the points. The primary purpose of a Hanan grid is to simplify the analysis of geometric properties of point sets.

A highly irregular graph typically refers to a graph that exhibits a significant degree of variation in some of its properties, such as vertex degrees, edge lengths, or connectivity. The term "irregular" can be used in various contexts, often in relation to specific characteristics of the graph. Here are a few interpretations: 1. **Irregular Degree Distribution**: In a graph, the degree of a vertex is the number of edges incident to it.

In graph theory, a **homogeneous graph** is a type of graph that exhibits uniformity in its structure regarding certain properties. The concept often refers to graphs where the connections or relationships between vertices have a certain degree of consistency or symmetry throughout the graph. One common context in which the term "homogeneous graph" is used is in the study of **homogeneous structures** in model theory.

A Kronecker graph is a type of random graph generated using the Kronecker product of matrices. It is a widely used model for generating large and complex networks, characterized by self-similarity and scale-free properties. The key idea behind a Kronecker graph is to recursively generate the adjacency matrix of the graph via a specific base matrix. ### Construction of Kronecker Graph 1.

A lattice graph is a type of mathematical graph that represents the structure of a lattice, which is a partially ordered set in which any two elements have a unique supremum (least upper bound) and an infimum (greatest lower bound). In simpler terms, a lattice graph visually depicts relationships between elements in a lattice structure. ### Characteristics of a Lattice Graph: 1. **Vertices and Edges**: Each vertex in the graph represents an element of the lattice.

"Leaf power" can refer to different concepts depending on the context in which it is used. Below are a few interpretations: 1. **Botanical Context**: In the field of botany, "leaf power" might refer to the ability of leaves to perform photosynthesis, which converts sunlight into chemical energy. This process is essential for plant growth and oxygen production.

A linear forest, in a forestry or ecological context, refers to a narrow strip of trees and vegetation that typically follows a linear path, such as along a stream, road, or property boundary. These linear formations are often used for various purposes including: 1. **Wildlife Corridors:** Linear forests can serve as habitat corridors for wildlife, allowing animals to move between different areas of habitat without having to cross open or developed land.

A list of graphs categorized by their number of edges and vertices typically refers to a classification of various types of graphs based on the relationships and connections they contain. Here are some common types of graphs organized by their number of vertices (V) and edges (E): 1. **Simple Graphs**: - **Complete Graph (K_n)**: A graph in which there is an edge between every pair of distinct vertices.

A locally linear graph refers to a concept in data analysis and geometry, particularly in the context of manifold learning and dimensionality reduction. In simpler terms, it is a type of graphical representation that exhibits linear characteristics within small neighborhoods or regions, even if the overall structure of the data is nonlinear.

The Lévy family of graphs is a concept in the field of probability theory and statistics, particularly in the context of Lévy processes. A Lévy process is a type of stochastic process that generalizes random walks and is characterized by stationary increments and continuity in probability. In particular, the Lévy family of graphs refers to the collection of parametric forms that describe the characteristic functions (or Laplace transforms) of Lévy processes.

A "map graph" typically refers to a graphical representation of geographical data where features, relationships, or various types of information are represented on a map. This term is often used in different contexts, including: 1. **Geographic Information Systems (GIS)**: Map graphs in GIS display spatial data, allowing users to visualize and analyze geographical relationships. These maps can represent various data types, like population density, weather patterns, or resource distribution.

A Meyniel graph is a specific type of graph in graph theory that is defined in relation to the properties of certain types of vertices. More formally, a Meyniel graph is one where the maximum degree of any vertex is at most one more than the average degree of the graph. Meyniel graphs are significant in various fields of combinatorics and graph theory, especially in discussions about certain classes of graphs and their properties.

In graph theory, a modular graph is a concept related to the idea of module or modularity in the context of substructures of a graph. The term "modular graph" can sometimes be used in discussions of modular decomposition, which is a technique for breaking down a graph into simpler components based on the concept of modules.

A **multipartite graph** is a specific type of graph used in graph theory, where the vertex set can be divided into multiple distinct subsets such that no two vertices within the same subset are adjacent. In other words, the edges of the graph only connect vertices from different subsets.

An outerplanar graph is a type of graph in which all of its vertices can be placed on the outer face of a planar drawing without any edges crossing. In other words, it can be drawn in such a way that all vertices are located on the boundary of the outer face, and no edges intersect except at their endpoints. Key characteristics of outerplanar graphs include: 1. **Planarity**: Outerplanar graphs are a subset of planar graphs.

In graph theory, an **overfull graph** typically refers to a graph that exceeds certain constraints, most commonly in the context of the vertex degrees or edge counts relative to some theoretical upper bound. The exact definition can vary based on the specific situation or properties being studied.

A Pairwise Compatibility Graph (PCG) is a type of graph that is used to represent the compatibility relationships between a set of items, entities, or individuals in various fields, such as computer science, biology, and social sciences. In a pairwise compatibility graph, the nodes (or vertices) represent the items, and the edges represent a compatibility relationship between pairs of items.

Panconnectivity refers to the concept of a highly interconnected and integrated network or system, where multiple devices, systems, or networks are seamlessly linked together. This idea is particularly relevant in the context of the Internet of Things (IoT), smart cities, and advanced communication technologies. In a panconnectivity environment, various technologies such as broadband internet, wireless communication, and sensor networks are utilized to promote interoperability among devices and services.

A parity graph is a type of graph that is primarily focused on the concept of parity, which pertains to whether the count of certain elements (like vertices or edges) is even or odd. In the context of graph theory, parity graphs can be understood in various ways depending on the specific problem or application being addressed. One common interpretation of parity graphs involves **vertex parity**.

A **planar graph** is a type of graph that can be embedded in the plane, meaning that it can be drawn on a flat surface such that its edges intersect only at their endpoints (vertices) and do not cross each other. In other words, a graph is planar if it can be represented in such a way that no two edges overlap except at their endpoints.

A Ptolemaic graph refers to a specific type of graph that is associated with a classical result in geometry known as Ptolemy's theorem. The theorem, attributed to the ancient Greek mathematician Ptolemy, pertains to cyclic quadrilaterals, which are four-sided figures where all vertices lie on a single circle.

A **quasi-bipartite graph** is a type of graph that is similar to a bipartite graph but with a relaxed condition. In a bipartite graph, the vertices can be divided into two disjoint sets such that no two vertices within the same set are adjacent. This means that edges only connect vertices from one set to those in the other set.

A scale-free network is a type of network characterized by a particular property in its degree distribution. In such networks, the distribution of connections (or edges) among the nodes follows a power law, which means that a few nodes (often referred to as "hubs") have a very high number of connections, while the majority of nodes have relatively few connections.

A self-complementary graph is a type of graph that is isomorphic to its own complement. In graph theory, for a given graph \( G \), the complement graph \( \overline{G} \) is formed by taking the same vertex set as \( G \) but including only those edges that are not present in \( G \).

A **split graph** is a type of graph in which the vertex set can be partitioned into two disjoint subsets: one subset forms a complete graph (often called the **clique**) and the other subset forms an independent set (meaning no two vertices in this subset have an edge between them). To summarize: - **Clique**: A subset of vertices such that every two vertices in this subset are connected by an edge.

As of my last knowledge update in October 2023, Squaregraph is a tool designed for businesses and organizations to help visualize data and create high-quality graphics for reporting, presentations, or marketing. It often aims to make data analysis more accessible and engaging through various visualization techniques, such as graphs, charts, and infographics. Squaregraph typically offers features that allow users to analyze data, generate visual content, and customize layouts to suit specific branding or informational needs.

A **strangulated graph** is a concept in graph theory that refers to a specific type of graph structure characterized by certain properties that relate to connectivity and edge restrictions. In particular, a graph is said to be strangulated if it has a partitioning of its vertex set into two subsets such that all vertices in one subset have a fixed degree (typically a very low degree) while vertices in the other subset have a much higher degree.

A **strongly chordal graph** is a specific type of graph that combines properties of both chordal graphs and certain restrictions on the structure of its cliques. 1. **Chordal Graph**: A graph is defined as chordal (or "circular" or "perfectly triangulated") if every cycle of four or more vertices has a chord. A chord is an edge that is not part of the cycle but connects two vertices of the cycle.

A threshold graph is a specific type of directed graph used in mathematics and computer science, particularly in the study of networks, social networks, and combinatorial optimization. It has a particular structure characterized by certain properties: 1. **Vertex Set**: A threshold graph is defined on a finite set of vertices. 2. **Edge Set**: The edges in a threshold graph are determined by a threshold value.

A **triangle-free graph** is a type of graph that does not contain any cycles of length three, which means it does not have any set of three vertices that are mutually connected by edges. In other words, if you pick any three vertices in the graph, at least one pair of those vertices will not be directly connected by an edge. Triangle-free graphs can be characterized using graph theory, and they have significant implications in various areas, including combinatorics, algorithm design, and social networks.

A trivially perfect graph is a special type of graph characterized by its cliques and independent sets. Specifically, a graph \( G \) is defined as trivially perfect if every induced subgraph of \( G \) has a clique that is also a maximum independent set.

A universal graph is a type of graph that contains all possible graphs of a certain type as subgraphs. More formally, a universal graph for a particular set of labeled graphs is a graph that includes every graph (or every isomorphism class of graphs) on a fixed number of vertices as a subgraph. For example, one well-known concept is the universal graph for finite graphs, which can contain all possible simple graphs on a finite set of vertices.

A graph is said to be **well-covered** if all of its maximal independent sets are of the same size. An independent set of a graph is a set of vertices no two of which are adjacent. A maximal independent set is an independent set that cannot be extended by including any adjacent vertex.

A **word-representable graph** is a type of graph that can be represented using words in such a way that the vertices of the graph correspond to distinct letters in a set of words, and an edge exists between two vertices if and only if the corresponding letters appear together in at least one of the words.

Individual graphs, in a general context, typically refer to graphical representations of data or information for single sets of variables or individual data points. Here are some relevant aspects: 1. **Single Variable Representation**: Individual graphs often display data related to a single variable, showcasing trends, distributions, or patterns. Common types include line graphs, bar charts, and pie charts. 2. **Statistical Analysis**: In statistics, individual graphs might be used to represent individual data points or cases within a dataset.

The 120-cell, also known as a hexadecachoron or 120-cell polytope, is a four-dimensional polytope (or 4-polytope) in geometry. It is one of the six regular convex 4-polytopes, which are the four-dimensional analogs of the three-dimensional Platonic solids. Here are some key properties of the 120-cell: 1. **Vertices**: The 120-cell has 600 vertices.

The 26-fullerene graph refers to a specific type of fullerene, which is a molecular structure made entirely of carbon, forming a hollow sphere, ellipsoid, or tube. Fullerenes are characterized by their spherical shapes, with carbon atoms arranged at the vertices of polygons, typically pentagons and hexagons. A 26-fullerene, specifically, is a fullerene that contains 26 carbon atoms.

The Berlekamp–Van Lint–Seidel graph (often abbreviated as BVS graph) is a specific type of strongly regular graph named after its contributors, Elwyn Berlekamp, Alexander Van Lint, and Franz Seidel. This graph is notable in combinatorial design theory and graph theory.

The Brouwer–Haemers graph is a specific type of graph in the field of graph theory. It is known for its interesting properties, particularly in relation to graph representations and properties of strongly regular graphs. The Brouwer–Haemers graph is a strongly regular graph with parameters \( (n, k, \lambda, \mu) = (12, 6, 2, 2) \), where: - \( n \) is the total number of vertices.

A "bull graph" may refer to a representation of a bullish trend in a financial market context, where the prices of assets are generally on the rise. Although "bull graph" is not a widely recognized term, the concept can be understood within the framework of bullish charts or graphs used in technical analysis. In financial markets, a bull market is characterized by rising prices, optimism, and investor confidence.

A butterfly graph is a type of network graph that resembles the shape of a butterfly when visualized. It is often used to represent parallel computations in computer science, particularly in the context of networks and interconnection systems. The butterfly graph has specific properties that make it useful for various applications, including: 1. **Structure**: A butterfly graph is typically defined recursively, meaning that it is constructed in layers.

A diamond graph is a specific type of graph in graph theory, characterized by its structure resembling a diamond shape. Formally, a diamond graph is a type of bipartite graph, denoted as \( K_{2,2} \), which consists of two disjoint sets of vertices with two vertices in each set, and each vertex from one set is connected to both vertices in the other set.

A dodecahedron is a three-dimensional geometric shape that is one of the five Platonic solids. It is characterized by having twelve flat faces, each of which is a regular pentagon. The dodecahedron has 20 vertices and 30 edges. In addition to its mathematical properties, dodecahedra can be found in various contexts, including architecture, art, and games (such as the shape of a 12-sided die often used in tabletop role-playing games).

The Errera graph is a specific type of directed graph used primarily in the study of graph theory and combinatorics. Named after the mathematician Jean Errera, it serves as a counterexample in certain contexts, particularly in discussions about graph properties like connectivity, cycles, and path lengths. The Errera graph has the following characteristics: - It consists of 3 vertices and is structured in a particular way. - The graph is directed, meaning that the edges have a direction associated with them.

The Goldner–Harary graph is a specific type of graph in the field of graph theory. It is notable for being a particularly well-studied example of a cubic graph (a graph where every vertex has degree 3) that is also a spanning subgraph of the complete graph \( K_6 \). The Goldner–Harary graph has the following properties: 1. **Vertices and Edges**: It contains 7 vertices and 21 edges.

A Golomb graph is a specific type of graph associated with the Golomb ruler, which is a set of markings at integer positions along an imaginary ruler such that no two pairs of markings have the same distance between them. In terms of graph theory, the Golomb graph is derived from the properties of such rulers. In a Golomb graph, each marking on the ruler corresponds to a vertex in the graph.

The Grötzsch graph is an important example in graph theory, particularly known for being a minimal example of a triangle-free graph that does not have a 3-coloring. The graph was named after the German mathematician Kurt Grötzsch, who constructed it in 1959. Here are some key characteristics of the Grötzsch graph: 1. **Vertices and Edges**: The Grötzsch graph consists of 11 vertices and 20 edges.

An icosahedron is a three-dimensional geometric shape that is one of the five Platonic solids. It is characterized by having 20 triangular faces, with three faces meeting at each vertex. The icosahedron is symmetric, meaning it has identical face and vertex configurations, and it has 12 vertices and 30 edges.

The Kittell graph, also known as the Kittell–Johnson graph, is a specific type of graph in graph theory. It is notable for its properties and structure, particularly in relation to its applications in combinatorial designs and algebraic constructions. Some of the key features of the Kittell graph include: - **Vertices and Edges:** The vertices of the graph represent certain combinatorial objects, and the edges depict specific relationships or interactions between these objects.