Questions on Graph theory

Question

I am in trouble with two problems. Thank you for your positive answer.

1. Is the incidence matrix of a directed graph totally unimodular?
2. Let bipartite graph without isolated vertices. Dose the maximum cardinality of a stable set equal the minuimum cardinality of an edge cover?

Answer 1

Hint:

• For the first question, incidence matrix yes (e.g. see this pdf). Do not confuse with adjacency matrix, for example $$\det\left[\begin{array}{ccc}0&1&1\\1&0&1\\1&1&0\end{array} \right] = 2.$$
• For the second question, see the König's theorem and consider the complements.

I hope this helps $\ddot\smile$