Sanjoy Nath's Geometric Junction Theory meets Sanjoy Nath's Geometrifying Trigonometry

 What are the most important graph theory algorithms? Show drafts Graph theory is a fundamental branch of mathematics with numerous applications in computer science and other fields. Here are some of the most important graph theory algorithms, categorized based on their functionalities: Traversal Algorithms: Depth-First Search (DFS): Systematically explores a graph by going as deep as possible on one branch before backtracking and exploring other branches. Used for finding connected components, topological sorting in Directed Acyclic Graphs (DAGs), and cycle detection. Breadth-First Search (BFS): Explores a graph layer-by-layer, visiting all neighbors of a node before moving to the next level. Used for finding shortest paths in unweighted graphs, finding connected components, and minimum spanning trees. Shortest Path Algorithms: Dijkstra's Algorithm: Finds the shortest path between a single source node and all other nodes in a weighted graph, considering positive edge weights. Bell...

 What are the counting problems related in Graph Theory? edit Show drafts volume_up Counting problems are a fundamental aspect of graph theory. They involve determining the number of graphs (or subgraphs) that satisfy certain properties. Here are some common types of counting problems in graph theory: Number of labeled graphs: This involves counting the number of distinct graphs with a given number of vertices (and possibly edges) where the order of the vertices matters. A classic example is the formula for the number of labeled graphs with n vertices, which is  2^(n(n-1)/2). Number of unlabeled graphs: Here, we only care about the structure of the graph, not the specific labeling of the vertices. Counting unlabeled graphs is generally more challenging than labeled graphs. Number of graphs with specific properties: This involves counting graphs that satisfy certain conditions, such as having a specific degree sequence (number of edges connected to each vertex), being acyclic (...