The Euler number is a topological invariant

1.1. Let \(X\) be a connected finite graph with \(V\) vertices and \(E\) edges.

1.2. Choose a maximal spanning tree \(T \subseteq X\).

1.3. Because \(T\) is a tree with \(V\) vertices, it has exactly \(V - 1\) edges and is contractible.

1.4. The quotient space \(X / T\) is homeomorphic to a wedge sum of \(k\) circles:\[X / T \cong \bigvee_{j=1}^k S^1,\]where the number of circles equals the number of remaining edges \(E - (V - 1) = E - V + 1 = 1 - \chi(X)\).

1.5. Because \((X, T)\) is a CW pair with contractible subcomplex \(T\), the quotient projection \(q: X \to X / T\) is a homotopy equivalence (Theorem 84.1).

1.6. Thus the fundamental group of \(X\) is a free group of rank \(k\):\[\pi_1(X, x_0) \cong \pi_1\left(\bigvee_{j=1}^k S^1\right) \cong F_k.\]

1.7. The rank \(k\) of a finitely generated free group is uniquely determined by its abelianization \(\pi_1(X)^{\text{ab}} \cong \mathbb{Z}^k\) (or first homology \(H_1(X) \cong \mathbb{Z}^k\)).

1.8. Expressing \(\chi(X)\) in terms of the rank gives:\[\chi(X) = 1 - k = 1 - \operatorname{rank}(\pi_1(X, x_0)).\]

1.9. Because the fundamental group is a topological (and homotopy) invariant, \(\chi(X)\) is a topological invariant for any connected finite graph \(X\).

2.1. Let \(X\) be a finite graph with \(c\) connected components \(X_1, \dots, X_c\).

2.2. The vertex and edge counts satisfy \(V = \sum_{i=1}^c V_i\) and \(E = \sum_{i=1}^c E_i\), so:\[\chi(X) = V - E = \sum_{i=1}^c (V_i - E_i) = \sum_{i=1}^c \chi(X_i).\]

2.3. Applying 1 to each connected component \(X_i\):\[\chi(X) = \sum_{i=1}^c \big(1 - \operatorname{rank}(\pi_1(X_i))\big) = c - \sum_{i=1}^c \operatorname{rank}(\pi_1(X_i)).\]

2.4. The number of connected components \(c\) and the fundamental groups of the individual components are all topological invariants of the space \(X\).