Do nearly balanced multigraphs have more spanning trees?

Let di the degree of the ith vertex of a mutigraph and λij be the number of edges between vertex i and vertex j. A multigraph is called nearly balanced if |di − di| ≤1 for all i ≠i′ and |λij −λij| for all i and all j,j′ Let be the collection of all the multigraphs with v vertices and e edges. It is shown that for any v, there is an e* such that if, then any nearly balanced graph in ζv, e has more spanning trees than any non-nearly-balanced graph in ζv, e.