Using mincuts to design secret sharing schemes in graphs

In this paper we propose a linear-time algorithm for determining the number of spanning trees in cographs; we derive formula for the number of spanning trees of a cograph G on n vertices and m edges, and prove that the problem of counting the number of spanning trees of G can be solved in O(n + m) times. Our proofs are based on the Kirchhoff matrix tree theorem which expresses the number of spanning trees of a graph as a function of the determinant of a matrix that can be easily construct from the adjacency relation of the graph. Our results generalize previous results regarding the number of spanning trees.