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.

Speciation is the evolutionary process by which populations evolve to become distinct species. The history of speciation is a vast topic that encompasses various mechanisms, theories, and examples throughout the history of life on Earth. 1. **Early Theories**: The concept of speciation, while understood in a rudimentary way, was not formalized until the 19th century. Charles Darwin's theory of evolution by natural selection provided a framework for understanding how new species might arise.

InfiniteGraph is a graph database and analytics platform designed for storing and querying complex interconnected data. Developed by InfiniteGraph, which was founded by a team including key individuals from the fields of computer science and data management, the platform enables users to manage large-scale graph data and perform advanced analytics on it. Key features of InfiniteGraph include: 1. **Scalability**: It is built to handle large volumes of data and can scale horizontally across distributed computing environments.

JanusGraph is an open-source, distributed graph database designed to handle large-scale graph data and complex queries. It is built to support various use cases such as social networks, recommendation systems, and fraud detection. Here are some key features and characteristics of JanusGraph: 1. **Scalability**: JanusGraph is designed to scale horizontally, making it suitable for handling large datasets across multiple servers.

NebulaGraph is an open-source, distributed graph database designed to manage and process large-scale graph data efficiently. It's built to handle complex relationships and connections within data, making it ideal for scenarios that require managed interconnections, such as social networks, recommendation systems, fraud detection, and knowledge graphs.

Ontotext GraphDB is a graph database management system designed for storing, retrieving, and managing complex interconnected data. It is particularly optimized for handling RDF (Resource Description Framework) data, which is commonly used in semantic web and linked data applications. GraphDB supports SPARQL, a powerful query language specifically for querying RDF data.

TerminusDB is an open-source graph database and knowledge graph technology designed for managing complex data. It is built for applications that require a flexible schema, semantic data modeling, and version control. TerminusDB allows users to create, maintain, and query databases that can represent complex relationships between entities more naturally than traditional relational databases.

DOT is a plain text graph description language primarily used for representing directed and undirected graphs. It is part of the Graphviz software suite, which is an open-source graph visualization tool. DOT allows users to specify the nodes and edges of a graph in a simple syntax, making it easy to define graph structures programmatically.

The concept of a "bondage number" typically arises in the context of graph theory. Specifically, the bondage number of a graph is defined as the minimum number of edges that must be removed from the graph in order to make it impossible to maintain a dominating set—that is, a set of vertices such that every vertex in the graph is either in the dominating set or is adjacent to a vertex in the dominating set—of a certain size.

Closeness centrality is a measure used in network analysis to determine how central or important a particular node (vertex) is within a graph. The idea behind closeness centrality is that nodes that are closer to all other nodes in the network are more central than those that are farther away. This metric is particularly useful for understanding the efficiency of spreading information or resources through the network.

A "Friendly index set" isn't a standard term widely recognized in mathematics or related fields as of my last update. However, it might be a term or concept from a specific domain, such as computer science, game theory, or a niche area of mathematics.

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.

Time-sharing systems represent a significant evolution in computing, allowing multiple users to interact with a computer simultaneously. This concept emerged from the need for more efficient use of computing resources, which were, at the time, expensive and limited. ### Key Stages in the Evolution of Time-Sharing Systems: 1. **Early Computing (1950s)**: - Computers were large, expensive, and primarily used for batch processing.

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.

A Type-in program refers to a type of computer program where the user is expected to manually enter (or "type in") the code into a computer or programming environment. This practice was particularly popular in the early days of personal computing, especially in the 1980s and 1990s, when magazines and books would publish source code listings for users to type into their own computers. Type-in programs often serve as a way to teach programming concepts, allowing users to learn by doing.

Chordal completion, also known as chordal graph completion, refers to the process of transforming a given graph into a chordal graph. A chordal graph (or "perfect graph") is defined as a graph in which every cycle of four or more vertices has a chord, which is an edge that is not part of the cycle but connects two vertices of the cycle. The process of chordal completion involves adding edges to the original graph in such a way that the resulting graph becomes chordal.