### Incidence matrix of a digraph – Ex. 1.5.6

The incidence matrix of a digraph (as defined in Ex. 1.5.6) gives no information about the ends of a directed loop. Should it be defined for loopless digraphs only?

The blog of the book by J.A. Bondy and U.S.R. Murty

Adrian Bondysaid on September 26, 2011 at 20:21The definition is intended to apply to all digraphs, with or without loops. The tail of a loop contributes 1 and the head -1 to the matrix entry, and these contributions cancel out. With this definition, the column sums are always zero, whether or not there are loops. Also, the columns corresponding to a cycle are linearly dependent, which accords with the matroidal viewpoint.

One can recognize loops from the incidence matrix, but not where they are attached. So digraphs, unlike graphs, are not completely determined by their incidence matrices.