Spanning trees in complete bipartite graphs and resistance distance in nearly complete bipartite graphs

Abstract

Using the theory of electrical network, we first obtain a simple formula for the number of spanning trees of a complete bipartite graph containing a certain matching or a certain tree.Then we apply the effective resistance (i.e., resistance distance in graphs) to find a formula for the number of spanning trees in the nearly complete bipartite graph G(m, n, p) = Km,n − pK2 (p ≤ min{m, n}), which extends a recent result by Ye and Yan who obtained the effective resistances and the number of spanning trees in G(n, n, p). As a corollary, we obtain the Kirchhoff index of G(m, n, p) which extends a previous result by Shi and Chen.

On spanning tree edge denpendences of graphs

“…) by Lemma 2.2, we still need to two results of Ge and Dong in [27] which gives the number of spanning trees which contain two types of subgraphs of a complete bipartite graph G = K r,s . For a graph G and a subgraph H of G, we use τ G (H) to denote the number of spanning trees of G containing H. The first result concerns the number of spanning tree of K r,s that contains a given subgraph tree T of K r,s .…”

“…Since the pioneering work of Cayley [3] who first determined the number of spanning trees of complete graphs, the number of spanning trees has been computed for various interesting families of graphs. For example, the number of spanning trees have been computed for complete bipartite graphs [4,5], complete multipartite graphs [6,7], cubic cycle C 3 N and the quadruple cycle C 4 N [9], graphs formed from a complete graph by deleting branches forming disjoint K-partite subgraphs [8], multi-star related graphs [10], K n -complements of quasi-threshold graphs [11], circulant graphs [13][14][15][16][17][18], K n -complements of asteroidal graphs [19], graphs with rotational symmetry [20], K m n ± G graphs [12], irregular line graphs [21] and line graphs [22,23], self-similar fractal models [24], 2-separable networks [25], a type of generalized Farey graphs [26], nearly complete bipartite graphs [27], Bruhat graph of the symmetric group [28], and so on.…”

“…[27] Let T be any tree which is a subgraph of K r,s . Thenτ Kr,s (T ) = (ms + nr − mn)r s−n−1 s r−m−1 , (2.1) where m = |V (T ) X|, n = |V (T ) Y |, and (X, Y ) is the bipartition of K r,s , with |X| = r, and |Y | = s.…”

Counting spanning trees in a complete bipartite graph which contain a given spanning forest

“…It turns out that this question is much harder than the case of complete graphs. In [3], this question was partially answered for two special cases: F is a matching or a tree plus several possible isolated vertices. In this paper, we obtain an explicit expression for τ F (K m,n ) for an arbitrary spanning forest F of K m,n .…”

Express the number of spanning trees in term of degrees

“…For the line graph G = L(H) of an arbitrary connected graph H, a relation between τ (G) and spanning trees in H was also established [3]. More works on τ (G) can be found in [5,6,10,11,15].…”