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

TL;DR

This paper introduces a novel approach to spanning tree enumeration by analyzing Laplacian matrices of special 2-threshold graphs through triangular rank-one perturbations, providing new proofs and characterizations.

Contribution

It offers a new characterization of special 2-threshold graphs and a spanning tree enumeration formula that generalizes existing results for threshold and Ferrers graphs.

Findings

New proofs for spanning tree enumeration formulas

Characterization of special 2-threshold graphs via Laplacian perturbations

Unified enumeration formula for threshold, Ferrers, and special 2-threshold graphs

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.