# Incidence matrix of a graph

Do incidence matrices fully characterize a graph? In other words, is it sufficient to retain the incidence matrix of a graph in order to know everything about it, namely, its connectivity (edge set), vertex set and underlying topology? As opposed to e.g. the degree matrix, which does not contain complete information on how the nodes are wired or directed in case of a digraph.

• Yes, each non-zero entry corresponds to a (vertex, edge) pair in the graph so the graph is fully characterized. I am not sure exactly what you mean by the "underlying topology," though. Commented Oct 23, 2019 at 14:17

If the graph has loops, then the incidence matrix does not determine it. We can tell that an edge is a loop, since the corresponding column is all $$0$$'s, but we can tell which vertex it is incident on. It would be easy to fix this, by just making the relevant entry $$1$$ instead of $$0$$ say, but this is not the usual way of defining the incidence matrix, so far as I am aware.