Hatcher-Algebraic-Topology · Problem 1.A.3

The Fundamental Group

Mathematics — Graph Theory · graphs-and-free-groups

Problem

For a finite graph XX define the Euler characteristic χ(X)\chi(X) to be the number of vertices minus the number of edges. Show that χ(X)=1\chi(X) = 1 if XX is a tree, and that the rank (number of elements in a basis) of π1(X)\pi_1(X) is 1−χ(X)1 - \chi(X) if XX is connected.

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