Counting rooted spanning forests in complete multipartite graphs

Jin and Liu discovered an elegant formula for the number of rooted spanning forests in the complete bipartite graph a1;a2 , with b1 roots in the rst vertex class and b2 roots in the second vertex class. We give a simple proof to their formula, and a generalization for complete m-partite graphs, using the multivariate Lagrange inverse. Y. Jin and C. Liu [3] give a formula for f(m; l;n; k), the number of spanning forests of the labelled complete bipartite graph Kn;m, where in the forest every tree is rooted, there are k roots in the rst vertex class (among the n vertices) and l roots in the second vertex class (among the m vertices), and the trees in the forest are not ordered. They discovered the elegant formula f(m; l;n; k) = m l n k n l m k (km+ ln lk): (1) The goal of the present note is generalization of (1) from complete bipartite to complete multipartite graphs, through a simple proof using the multivariate Lagrange inverse. Let f(a1; b1; :::; am; bm) denote the number of spanning forests of the labelled complete multipartite graphKa1;a2;:::;am , where in the forest every tree is rooted, there are bi roots in the i th vertex class for i = 1; 2; :::;m, and the trees in the forest are not ordered. Let wi(t1; :::; tm) denote the multivariate exponential generating function (EGF) of the numbers f(a1; 0; :::; ai; 1; :::; am; 0) (the number of rooted spanning trees of the complete multipartite graph Ka1;a2;:::;am , if the root has to be in the i class), i.e. wi(t1; :::; tm) = 1 X a1=0 ::: 1 X ai=1 ::: 1 X am=0 f(a1; 0; :::; ai; 1; :::; am; 0) m Y k=1 tk k ak! : (2) Research partially supported by the NSF Grant 0072187 and the Hungarian NSF Grant T 032455. The key identity for our argument is tie (w1+w2+:::+wm) wi = wi for i = 1; 2; :::;m: (3) The proof of formula (3) is based on the following combinatorial decomposition. Given a rooted spanning tree of the complete multipartite graph Ka1;a2;:::;am , where the root is in the i class, remove the root vertex from the tree to obtain a spanning forest of Ka1;a2;:::;ai 1;:::;am , and mark the former neighbors of the eliminated root vertex as roots in the forest. This decomposition establishes a bijection between the following two sets: the set of rooted spanning trees of the complete multipartite graph Ka1;a2;:::;am , where the root is in the i vertex class, and the set of some ordered pairs, where the rst entry of the ordered pair is one of the vertices of the i vertex class, the second element of the ordered pair is a rooted spanning forest of Ka1;a2;:::;ai 1;:::;am , where the vertex from the rst entry is removed from the i vertex class, and the trees of the forest are not ordered. Now tie (w1+w2+:::+wm) wi is the EGF of the set of ordered pairs in question, according to the Exponential Formula; and wi is the same EGF by the bijection. Set i(w1; w2; :::; wm) = e (w1+w2+:::+wm) wi . According to the multiplication rule of EGF's, Qm k=1w bk k is the multivariate exponential generating function of the number of rooted spanning forests of complete m-partite graphs, with bk roots in the k th vertex class, where the trees rooted in the same part are ordered; hence f(a1; b1; :::; am; bm) = a1!a2! am! b1!b2! bm! [t1 1 t a2 2 t am m ] m Y