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Spanning trees in complete bipartite graphs and resistance distance in nearly complete bipartite graphs
Citation
Ge, J., & Dong, F. (2020). Spanning trees in complete bipartite graphs and resistance distance in nearly complete bipartite graphs. Discrete Applied Mathematics, 283, 542-554. https://doi.org/10.1016/j.dam.2020.02.002
Abstract
Using the theory of electrical network, we first obtain simple formulas for the number of spanning trees of a complete bipartite graph containing a certain matching or a certain tree. Then we compute the effective resistances (i.e., resistance distance in graphs) in the nearly complete bipartite graph G(m,n,p) = Km,n -pK2 (p≤ min｛m,n｝which extends a recent result (Ye and Yan, 2019) on the effective resistances 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. Using the effective resistances in G(m,n,p), we find a formula for the number of spanning trees of G(m,n,p). In the end, we prove a general result for the number of spanning trees of a complete bipartite graph containing several edges in a certain matching and avoiding others.
Date Issued
2020
Publisher
Elsevier
Journal
Discrete Applied Mathematics
Grant ID
NSFC (Grant no. 11701401)
China Scholarship Council (Grant no. 201708515087)