双方形图的$K_n$$补集中生成树的数目

Pub Date : 2024-06-01 DOI:10.1007/s10801-024-01341-y

Helin Gong, Yu Gong, Jun Ge

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引用次数: 0

摘要

对于完整图 $K_n$的子图 G，G 的 $K_n$-补集（用 $K_n-G$表示）是从 $K_n-G$中删除 G 的所有边而得到的图。在本文中，我们用 G 的所有诱导平衡双方子图的双向矩阵的行列式来表示双方子图 G 的 $K_n$-complement $K_n-G$的生成树数，这些矩阵都是非奇异的，我们还推导出了各种重要类别的双方子图 G 的 $K_n$-complement $K_n-G$的生成树数公式，其中一些公式概括了之前的一些结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

The number of spanning trees in $$K_n$$ -complement of a bipartite graph

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The number of spanning trees in $$K_n$$ -complement of a bipartite graph

For a subgraph G of a complete graph $K_n$, the $K_n$-complement of G, denoted by $K_n-G$, is the graph obtained from $K_n-G$ by removing all the edges of G. In this paper, we express the number of spanning trees of the $K_n$-complement $K_n-G$ of a bipartite graph G in terms of the determinant of the biadjcency matrices of all induced balanced bipartite subgraphs of G, which are nonsingular, and we derive formulas of the number of spanning trees of $K_n-G$ for various important classes of bipartite graphs G, some of which generalize some previous results.

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