Graph families

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.