In graph theory, extensions and generalizations of graphs refer to various constructs and modifications of standard graph representations, allowing for additional features or alternative interpretations. Here are some common concepts related to extensions and generalizations of graphs: ### Extensions of Graphs 1. **Subgraphs**: A subgraph is formed by a subset of the vertices and edges of a graph. It retains some or all of the connections present in the original graph.
Extremal graph theory is a branch of combinatorial mathematics that studies the extremal properties of graphs. Specifically, it focuses on questions related to the maximal or minimal number of edges in a graph that satisfies certain properties or conditions. The primary goal is often to determine the extremal (that is, maximum or minimum) values for specific parameters of graphs (like the number of edges, number of vertices, etc.) that meet certain constraints, such as containing a particular subgraph or avoiding certain configurations.
Graph databases are a type of database specifically designed to represent and store data in the form of graphs, which consist of nodes (entities) and edges (relationships). This model excels in scenarios where relationships and connections between data points are crucial and often complex. ### Key Characteristics of Graph Databases: 1. **Nodes and Edges**: - **Nodes**: Represent entities or objects, such as people, places, products, etc.
The term "operator system" can refer to different concepts depending on the context. Here are a few interpretations: 1. **Mathematical Operator Systems**: In mathematics, particularly in functional analysis and operator algebra, an operator system is a certain type of self-adjoint space of operators on a Hilbert space that has a structure similar to that of a C*-algebra but is more general.
Graph Description Languages (GDLs) are specialized languages used to specify, represent, and manipulate graphs or graph-like structures. These languages provide a way to express the nodes, edges, properties, and relationships of graphs in a formal manner, making it easier for software tools and algorithms to process and analyze graph data. **Key Features of Graph Description Languages:** 1.
Graph theory is a rich area of mathematics with many interesting unsolved problems. Here are some notable ones: 1. **Graph Isomorphism Problem**: This problem asks whether two finite graphs are isomorphic, meaning they have the same structure regardless of the labels of their vertices. While there are polynomial-time algorithms for certain classes of graphs, a general polynomial-time solution for all graphs remains elusive.
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 2. You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website.
Figure 3. Visual Studio Code extension installation.
Figure 5. . You can also edit articles on the Web editor without installing anything locally. Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension. 
- Infinitely deep tables of contents:
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact