Example: The First Betti Number in Graph Theory
In topological graph theory the first Betti number of a graph G with n vertices, m edges and k connected components equals
This may be proved straightforwardly by mathematical induction on the number of edges. A new edge either increments the number of 1-cycles or decrements the number of connected components.
See cyclomatic complexity for an application of the first Betti number in software engineering.
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