A new technique for the characterization of graphs with a maximum number of spanning trees Original Research Article

Let Γ(n,e) denote the class of all simple graphs on n nodes and e edges. The number of spanning trees of a graph G is denoted by t(G). A graph G0∈Γ(n,e) is said to be t-optimal if t(G0)⩾t(G) for all G∈Γ(n,e). The problem of characterizing t-optimal graphs for arbitrary n and e is still open, although characterizations of t-optimal graphs for specific pairs (n,e) are known. We introduce a new technique for the characterization of t-optimal graphs, based on an upper bound for the number of spanning trees of a graph G in terms of the degree sequence and the number of induced paths of length two of the complement of G. The technique yields the following new results:<