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arXiv · 2103.13596

Spanning tree enumeration via triangular rank-one perturbations of graph Laplacians

Christian Go·Zhong Xuan Khwa·Xinyu Luo·Matthew T. Stamps
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Abstract

We present new short proofs of known spanning tree enumeration formulae for threshold and Ferrers graphs by showing that the Laplacian matrices of such graphs admit triangular rank-one perturbations. We then characterize the set of graphs whose Laplacian matrices admit triangular rank-one perturbations as the class of special 2-threshold graphs, introduced by Hung, Kloks, and Villaamil. Our work introduces (1) a new characterization of special 2-threshold graphs that generalizes the characterization of threshold graphs in terms of isolated and dominating vertices, and (2) a spanning tree enumeration formula for special 2-threshold graphs that reduces to the aforementioned formulae for threshold and Ferrers graphs. We consider both unweighted and weighted spanning tree enumeration.

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Christian Go, Zhong Xuan Khwa, Xinyu Luo, Matthew T. Stamps. 2021-03-25. Spanning tree enumeration via triangular rank-one perturbations of graph Laplacians. https://arxiv.org/abs/2103.13596

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