Minimum spanning tree theory has a wide application in many fields. But in many practical problems, we are often faced with the heterogeneous node weighted graph with both edge weight and node weight be considered. In this paper, we present the definition and the mathematical model of the best spanning tree, then raise an algorithm of the best spanning tree, finally, prove that the algorithm is...

Given a weighted graph, let W 1 ; W 2 ; W 3 ; : : : denote the increasing sequence of all possible distinct spanning tree weights. Settling a conjecture due to Kano, we prove that every spanning tree of weight W 1 is at most k ? 1 edge swaps away from some spanning tree of weight W k. Three other conjectures posed by Kano are proven for two special classes of graphs. Finally, we consider the al...