24. Graph Theory Basics
Graph theory encompasses a vast realm of concepts involving vertices and edges, including specialized structures like complete graphs, bipartite graphs, and cycle graphs. Fundamental theorems such as the Handshaking theorem and Euler's theorem provide insight into the properties of graphs, particularly regarding vertex degrees and connectivity. Understanding these concepts establishes a foundation for exploring more complex topics in graph theory.
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What we have learnt
- Graphs consist of vertices and edges, with types differentiated into directed and undirected graphs.
- Bipartite graphs require a partitioning of vertices where edges connect only between distinct subsets.
- Euler's theorem indicates that the number of vertices with odd degrees in an undirected graph is always even.
Key Concepts
- -- Graph
- A structure made up of vertices (nodes) connected by edges.
- -- Directed graph
- A graph where edges have a direction, indicated by ordered pairs of vertices.
- -- Undirected graph
- A graph where edges do not have a direction, represented by unordered pairs.
- -- Simple graph
- A graph with no self-loops and at most one edge between each pair of vertices.
- -- Degree of a vertex
- The number of edges incident to a vertex, counting self-loops twice.
- -- Bipartite graph
- A simple graph whose vertices can be divided into two distinct sets such that no two graph vertices within the same set are adjacent.
- -- Complete bipartite graph
- A bipartite graph where every vertex in one partition set is connected to every vertex in the other partition set.
- -- Euler's theorem
- A theorem stating that the number of vertices with an odd degree in an undirected graph is always even.
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