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 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 **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 **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 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.

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 **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 **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.

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.

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.