The problem studied is the following: Find a simple connected graph G with given numbers of vertices and edges which minimizes the number tμ(G), the number of spanning trees of the multigraph obtained from G by adding μ parallel edges between every pair of distinct vertices. If G is nearly complete (the number of edges qis ≥(2P)−p+2 where p is the number of vertices), then the solution to the minimization problem is unique (up to isomorphism) and the same for all values of μ. The present paper investigates the case whereq<(2P)−p+2. In this case the solution is not always unique and there does not always exist a common solution for all values of μ. A (small) class of graphs is given such that for any μ there exists a solution to the problem which is contained in this class. For μ = 0 there is only one graph in the class which solves the problem. This graph is described and the minimum value of t0(G) is found. In order to derive these results a representation theorem is proved for the cofactors of a special class of matrices which contains the tree matrices associated with graphs.
