I am reading Weibel, An Introduction to Homological Algebra. In chapter one (Exercise 1.1.6.), there is the definition of the homology of a graph, that I can't understand.
Let $v_1,\ldots,v_V$ be the vertices and $e_1,\ldots,e_E$ the edges of a finite graph. For any ring $R$, let $C$ be the $R$-module chain complex with $C_1$ the free $R$-module on the set of edges, $C_0$ the free $R$-module on the set of vertices, all others $C_n$ the zero $R$-module and the only non-zero differential $d\colon C_1\to C_0$, given by the incidence matrix.
Question: what does it mean that the incidence matrix of the graph works as the differential of the chain complex $C$?
The incidence matrix of a graph has a row for each vertex, a column for each edge, and a $0$ or $1$ in each entry depending on whether the corresponding vertex is on the corresponding edge. As an $V\times E$ matrix, it defines a linear transformation from $R^E$ to $R^V$.
This is just a fancy way of saying the differential of an edge is the formal sum of its endpoints.