Deletion-contraction for a unified Laplacian and applications

TL;DR

This paper introduces a unified graph Laplacian with vertex and edge weights, providing combinatorial interpretations, deletion-contraction relations, and applications to graph properties and Stanley's conjecture.

Contribution

It defines a new weighted Laplacian unifying classical variants, establishes deletion-contraction relations, and applies these to graph theory problems and conjectures.

Findings

Unified Laplacian with vertex and edge weights

Deletion-contraction relations for weighted Laplacian

Applications to graph cuts, independent sets, and Stanley's conjecture

Abstract

We define a graph Laplacian with vertex weights in addition to the more classical edge weights, which unifies the combinatorial Laplacian and the normalised Laplacian. Moreover, we give a combinatorial interpretation for the coefficients of the weighted Laplacian characteristic polynomial in terms of weighted spanning forests and use this to prove a deletion-contraction relation. We prove various interlacing theorems relating to deletion and contraction, as well as to rectangular tilings, drawing on the work of Brooks, Smith, Stone and Tutte on square tilings. Additionally, we show that the weighted Laplacian also satisfies a vertex analogue of deletion-contraction. We give applications of weighted Laplacian eigenvalues to sparse cuts, independent sets and graph colouring, and establish new cases of a conjecture of Stanley on distinguishing nonisomorphic trees.