Mathematics

Tutte's theorem asserts that the number of directed spanning trees, rooted at a particular vertex of a directed graph, is equal to the determinant of an appropriate reduction of the Laplacian matrix associated to this graph. I will finish a proof of this result that relies on a particular factorization of the Laplacian matrix, and state some extensions to weighted directed graphs. Finally, I will discuss some open problems related to the vulnerability or resilience of networks, motivated mainly by ecological networks, although generalizations to information or social networks are possible as well.