Graph operations

Graph operations refer to various manipulations and processes that can be performed on graphs, which are mathematical structures used to model pairwise relationships between objects. A graph consists of vertices (or nodes) and edges (connections between the vertices). Graph operations can help analyze, modify, or derive new graphs from existing ones. Here are some common types of graph operations: 1. **Graph Creation**: - **Adding Vertices**: Introducing new vertices to an existing graph.

Graph products are operations that combine two or more graphs to create a new graph, and these products can capture different structural relationships between the original graphs. There are several types of graph products, each with its own definition and properties.

The Cartesian product of two graphs \( G_1 = (V_1, E_1) \) and \( G_2 = (V_2, E_2) \) is a graph constructed by combining the vertices of the two graphs in a specific way.

In graph theory, the graph product is a way to combine two graphs to create a new graph. There are several types of graph products, each with different properties and applications.

The lexicographic product (or Cartesian product) of two graphs \( G = (V_G, E_G) \) and \( H = (V_H, E_H) \) is a graph denoted by \( G \cdot H \) (or sometimes \( G[H] \) or \( G \square H \)).

Min-plus matrix multiplication is an operation that extends the traditional concept of matrix multiplication using the minimum and plus operations instead of the standard addition and multiplication. Specifically, in min-plus algebra: 1. **Addition** in the min-plus context is defined as taking the minimum of two values. For example: \[ a \oplus b = \min(a, b) \] 2. **Multiplication** involves addition.

The modular product of graphs is a way to combine two graphs into a new one that captures certain structural properties of the original graphs. Specifically, it preserves the modularity of the vertex sets in each graph.

The rooted product of graphs is a specific operation that combines two graphs in a way that preserves some structural properties of both.

The strong product of two graphs \( G \) and \( H \), denoted as \( G \boxtimes H \), is a graph that combines the structures of both \( G \) and \( H \) in a way that incorporates features from both the Cartesian product and the tensor product of graphs.

The tensor product of graphs, also known as the Kronecker product or categorical product, is a way to combine two graphs into a new graph.

The zig-zag product is an operation on graphs, specifically useful in the field of combinatorial design and expander graphs. It allows the construction of a new graph from two existing graphs in a way that preserves certain properties, typically expanding size and connectivity characteristics. For two graphs \( G \) and \( H \): - Let \( G \) be a graph with vertex set \( V_G \) and \( H \) be a directed graph with vertex set \( V_H \).

A **bipartite double cover** of a graph is a specific type of covering graph that is particularly relevant in the context of bipartite graphs. To elaborate, consider the following concepts: 1. **Bipartite Graphs**: A bipartite graph is a graph that can be divided into two disjoint sets of vertices \( U \) and \( V \) such that every edge connects a vertex in \( U \) to a vertex in \( V \).

"Bipartite half" might refer to concepts within graph theory, particularly regarding bipartite graphs. A **bipartite graph** is a type of graph where the set of vertices can be divided into two distinct sets such that no two graph vertices within the same set are adjacent.

A **clique graph** is a concept in graph theory that pertains to representing cliques within a given graph. A **clique** in a graph is a subset of its vertices such that every two distinct vertices in the subset are adjacent, meaning there is an edge connecting each pair of vertices. In simpler terms, a clique is a complete subgraph.

A **cograph** is a type of graph that can be defined as a graph without any induced subgraphs that are isomorphic to a path of four vertices (also known as a **P4**). In simpler terms, cographs can be constructed using two operations: the **disjoint union** and the **join** (or clique-sum) of two graphs.

In graph theory, the complement of a graph is a graph that contains the same set of vertices but has edges that are not present in the original graph.

The disjoint union of graphs is a concept in graph theory that combines two or more graphs into a new graph in such a way that the original graphs do not share any vertices or edges. Here's how it works: 1. **Graphs Involved**: Suppose you have two or more graphs \( G_1, G_2, \ldots, G_n \).

Edge contraction is a fundamental operation in graph theory that involves modifying a given graph by merging two adjacent vertices into a single vertex. This operation is often used in various algorithms and theoretical contexts, such as simplifying graphs, network flow problems, and studying graph properties. Here's how edge contraction works: 1. **Consider an edge** \(e\) between two vertices \(u\) and \(v\) in a graph \(G\).

The Goldberg–Coxeter construction is a method used in geometry, particularly in the study of polyhedra and polyhedral structures. It provides a systematic way to generate a class of convex polyhedra, particularly those that can be described as geometric realizations of certain types of combinatorial structures known as "spherical polyhedra.

"Graph power" is not a standard term in mathematics or computer science, so it may refer to different concepts depending on the context. Here are some interpretations: 1. **Graph Theory**: In the context of graph theory, "power" can refer to the concept of a power of a graph, which is related to the construction of new graphs by connecting vertices based on paths of a certain length.

An **induced subgraph** is a concept from graph theory. Given a graph \( G = (V, E) \), where \( V \) is the set of vertices and \( E \) is the set of edges, an induced subgraph is formed by a subset of the vertices \( U \subseteq V \) along with all of the edges in \( E \) that have both endpoints in \( U \).

A line graph is a type of chart used to display information that changes over time. It consists of a series of data points, called "markers," connected by straight line segments. Line graphs are particularly useful for showing trends, patterns, and relationships between two variables. ### Key Features of Line Graphs: 1. **Axes**: - The horizontal axis (x-axis) typically represents the independent variable (often time).

A medial graph is a concept used in the field of computational geometry, particularly in the areas of shape analysis and mesh processing. It represents the intrinsic structure of a geometric shape or surface. In essence, the medial graph captures the topological and geometrical characteristics of the shape's skeleton or centerlines. It is achieved through various methods, which typically involve identifying points that are equidistant from the boundary of the shape.

A **moral graph** is a concept used in the fields of graph theory and probabilistic graphical models, particularly in the context of Bayesian networks and Markov networks. The moral graph is derived from a directed acyclic graph (DAG) representing a Bayesian network. ### How to Construct a Moral Graph: 1. **Start with a Directed Graph:** Begin with a Bayesian network, which is typically represented as a directed acyclic graph (DAG).

In mathematics and graph theory, a Mycielski graph, or Mycielski construction, is a method for constructing new graphs from existing ones. The Mycielski construction is used primarily to create triangle-free graphs, which are graphs that do not contain any cycles of length three (triangles). The construction works as follows: 1. **Start with a graph \( G \)**: This can be any simple graph.

A **quotient graph** is a concept in graph theory that arises when you take a graph and partition its vertices into equivalence classes, then construct a new graph where each equivalence class is represented as a single vertex. ### Key Components of a Quotient Graph: 1. **Original Graph (G)**: Start with a graph G = (V, E), where V is the set of vertices and E is the set of edges.

A series-parallel graph is a specific type of graph that can be constructed from a single edge by repeatedly applying two operations: series composition and parallel composition. These operations allow the building of more complex graphs while maintaining certain structural properties.

A Simplex graph is related to the concept of simplices in geometry and topology. In mathematical terms, a simplex is the generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. For example: - A 0-simplex is a single point. - A 1-simplex is a line segment connecting two points. - A 2-simplex is a triangle, defined by three points. - A 3-simplex is a tetrahedron, defined by four points.