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arXiv · 1904.07766

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

Jun Ge·Fengming Dong
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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 $G(m,n,p)=K_{m,n}-pK_2$ $(p\leq \min\{m,n\})$, 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.

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BibTeXRIS

Jun Ge, Fengming Dong. 2019-04-16. Spanning trees in complete bipartite graphs and resistance distance in nearly complete bipartite graphs. https://doi.org/10.1016/j.dam.2020.02.002

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