(m, N) - The Tree To Determine And Count

Tree is a basic concept in Graph Theory, and it was firstly extended in higher space in [2] by Beineke and Pippert. Later, it was developed further in n -dimensional complexes in [1] by Dewdney, and the concept of (m, n)-tree was obtained at the same time. Furthermore similar to the tree-characterization of Graph Theory, the following result about the (m, n)-tree characterization was given:If K is a (m, n)-tree, then K satisfies the following conditions:(1) K is (m, n) -connected;(2) K contains no (m, n) -circuits;(3) where k = 1,2,...n; Bm,n(k,K) = andαk(K) denotes the number of k -dimensional simplexes in K.In the paper, on the basis of [1] and [5], by combining the above three basic features, some new necessary and sufficient conditions of (m, n)-tree are firstly given .At the last , by the definition of (m, n)-tree in Graph Theory and using the combinatorial method, the following counting formula of labeled (m.n)-tree of order α0 was given:...