homotopy equivalence between graph and sphere and related Euler characteristic

Question

I saw a question here The relationship between the Euler characteristic and the fundamental group of finite connected graphs saying that :

Any connected graph is homotopy equivalent to wedge of $S_1$

I've got 3 questions:

1. Why? And for example, for a connected graph like a hexagon or $K_{3,3}$, how to write it as wedge of $S_1$?
2. Euler characteristic is a characteristic class on the vector bundle of a manifold. But for a graph, how can we define such a vector bundle on a graph which corresponds to its Euler characteristic class?
3. For cellular complex, can we view it as a graph and compute its topological invariants using topology of a graph?

Answer 1

1. Consider a maximal tree $T$ in a connected graph $G$, then the map $G \to G/T$ is a homotopy equivalence and $G/T$ is a wedge of one-spheres.

2. The Euler characteristic is defined in many contexts, you don't need to have vector bundles on a manifold. For a finite graph, the Euler characteristic is simply the number of vertices minus the number of edges.

3. No, in a general cellular complex you have higher-dimensional cells. A graph is a one-dimensional CW complex, because you have only cells in dimension $0$ (corresponding to vertices) and cells in dimension $1$ (corresponding to edges)