In graph theory, the **metric dimension** of a graph is a concept that relates to the ability to uniquely identify the vertices of the graph based on their distances to a specific set of vertices known as "experimenters" or "measuring points.

The Strahler number is a concept used in hydrology and geomorphology to describe the hierarchical order of a stream or river system. It provides a way to classify streams based on their drainage structure. The Strahler number is determined according to the following rules: 1. **Headwater Streams**: Any stream segment that has no tributaries is assigned a Strahler number of 1.

The Tutte–Grothendieck invariant is an important concept in graph theory and combinatorics, associated with the study of matroids and graphs. This invariant is commonly denoted as \( T(G) \) for a graph \( G \) and is defined in terms of the graph's structure, specifically its connected components and edges.

Bounded expansion is a concept in graph theory that pertains to the behavior of certain classes of graphs, particularly in relation to their structure and properties. A family of graphs is said to have bounded expansion if, roughly speaking, the density of the graphs in the family does not grow too quickly as the size of the graphs increases.

The **clique-sum** is a graph operation used primarily in graph theory, particularly in the study of graph properties and constructions. This operation allows you to combine two graphs in a way that preserves some of their characteristics while introducing new structure. Here’s how the clique-sum operates: 1. **Graphs Involved**: You start with two graphs, say \(G_1\) and \(G_2\).

"Shallow minor" is not a standard term widely recognized in music theory or other disciplines. However, the phrase could be interpreted in a few ways depending on the context in which it's used: 1. **Musical Context**: If we're discussing music, it might refer to a minor key (like A minor, B minor, etc.) that feels less intense or lacks depth, possibly due to its simplicity in composition or harmony.

"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.

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.

In graph theory, the term "core" refers to a specific type of subgraph that captures some essential structural properties of the original graph. A **core** of a graph is often defined as a maximal subgraph in which every vertex has a degree (number of edges connected to it) of at least \( k \). This means that for a \( k \)-core, every vertex in the graph has at least \( k \) connections.

Graceful labeling is a concept in graph theory related to labeling the vertices of a graph in a specific way that satisfies certain criteria. A graph is said to be gracefully labeled if it can be assigned labels (usually integers) to its vertices such that the following conditions are met: 1. The labels assigned to the vertices are distinct integers, typically taken from the set {0, 1, 2, ...

The term "induced path" typically arises in various contexts, such as in mathematics, particularly in graph theory. In graph theory, an **induced path** refers to a specific kind of subgraph of a graph.

In the context of graph theory, a **minimum cut** refers to a specific type of partitioning of a graph that separates the vertices into two distinct subsets while minimizing the total weight of the edges that cross between these two subsets. ### Key Concepts: - **Graph**: A collection of vertices (nodes) connected by edges (links). - **Cut**: A cut in a graph is a way of dividing the graph's vertices into two disjoint subsets.

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.

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 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.

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 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 Smith graph is a specific type of graph in graph theory. It is a 5-regular graph on 14 vertices, meaning that each vertex has exactly 5 edges connecting it to other vertices. The Smith graph has a unique property: it is both vertex-transitive and edge-transitive, which means that the structure looks the same from any vertex and from any edge. Certain characteristics of the Smith graph include: - It has 14 vertices and 35 edges.

Blockmodeling is a methodological approach used in social network analysis to simplify and analyze complex social networks by grouping nodes (typically individuals or organizations) into blocks based on their structural characteristics and relationships. The primary goal of blockmodeling is to reveal patterns and underlying structures within a network, making it easier to understand the relationships among actors.

Confirmatory blockmodeling is a statistical technique used in social network analysis to test hypothesized structures within network data. It is concerned with identifying and validating specific patterns of connections (or relationships) among a set of actors (nodes) that belong to different groups (blocks). This method is useful in understanding how these groups interact within a network.