The **incidence matrix** of a directed graph *D* is a *p* × *q* matrix where *p* and *q* are the number of vertices and edges respectively, such that if the edge leaves vertex, if it enters vertex and 0 otherwise (Note that many authors use the opposite sign convention.).

An **oriented incidence matrix** of an undirected graph *G* is the incidence matrix, in the sense of directed graphs, of any orientation of *G*. That is, in the column of edge *e*, there is one +1 in the row corresponding to one vertex of *e* and one −1 in the row corresponding to the other vertex of *e*, and all other rows have 0. All oriented incidence matrices of *G* differ only by negating some set of columns. In many uses, this is an insignificant difference, so one can speak of *the* oriented incidence matrix, even though that is technically incorrect.

The oriented or unoriented incidence matrix of a graph *G* is related to the adjacency matrix of its line graph *L*(*G*) by the following theorem:

where is the adjacency matrix of the line graph of *G*, *B*(*G*) is the incidence matrix, and is the identity matrix of dimension q.

The Kirchhoff matrix is obtained from the oriented incidence matrix *M*(*G*) by the formula

The integral cycle space of a graph is equal to the null space of its oriented incidence matrix, viewed as a matrix over the integers or real or complex numbers. The binary cycle space is the null space of its oriented or unoriented incidence matrix, viewed as a matrix over the two-element field.

Read more about Incidence Matrix: Incidence Structures

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