Graph theory studies vertices and the edges connecting them: which graph paths exist, how efficiently we can travel, and which connections are indispensable. A road map becomes a graph when junctions are vertices and road segments are edges; a communications network uses routers and links instead. The drawing helps us see connections, but its apparent lengths need not be the costs being optimized. Record travel times or transmission costs separately in a weighted graph.
The appeal of graph theory is that one model serves many practical questions. Graph paths describe routes, connected components of a graph identify isolated regions, and minimum spanning trees connect all locations cheaply. The shortest path problem asks a different question: find the cheapest journey between specified locations. Our four-vertex example below makes that distinction concrete and shows how the Dijkstra algorithm discovers a route without trying every graph path.
Graph theory is a branch of mathematics and computer science that studies the properties and applications of graphs. A graph is a collection of nodes (or vertices) connected by edges (or arcs). Graph theory provides a framework for modeling and analyzing relationships and interactions in various systems. Key concepts in graph theory include: 1. **Vertices and Edges**: The basic building blocks of a graph. Vertices represent entities, while edges represent the connections or relationships between them.
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