Graph invariants are properties or characteristics of a graph that remain unchanged under specific operations or transformations, such as isomorphisms (relabeling of vertices), graph expansions, or contractions. These invariants provide essential insights into the structure and behavior of graphs and are crucial in various fields, including mathematics, computer science, and network theory.
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.
The term "18th century in computing" can be somewhat misleading, as the 18th century (1701-1800) predates the invention of modern computers. However, this period was significant for laying the groundwork for later advancements in computing through developments in mathematics, logic, and mechanical devices.
The 21st century in computing is characterized by rapid advancements in technology and a significant transformation in how we interact with and utilize computers. Some key highlights of this era include: 1. **Internet and Connectivity**: The widespread adoption of the internet transformed computing, enabling global connectivity, access to vast amounts of information, and the rise of online services and social media platforms. 2. **Mobile Computing**: The proliferation of smartphones and tablets revolutionized personal computing.
Category theory is a branch of mathematics that focuses on the abstract study of mathematical structures and relationships between them. It provides a unifying framework to understand various mathematical concepts across different fields by focusing on the relationships (morphisms) between objects rather than the objects themselves. Here are some key concepts in category theory: 1. **Categories**: A category consists of objects and morphisms (arrows) that map between these objects. Each morphism has a source object and a target object.