Algebraic Properties of Generalized Graph Laplacians: Resistor Networks, Critical Groups, and Homological Algebra

TL;DR

This paper develops an algebraic framework unifying resistor networks, critical groups, and Laplacian eigenvalues using generalized graph Laplacians and homological algebra, providing new tools for analysis and simplification.

Contribution

It introduces a generalized critical group for graphs with boundary and relates it to homological algebra, offering new methods for analyzing graph Laplacians and resistor networks.

Findings

Defines a generalized critical group using homological algebra.

Provides an algorithm for simplifying computation of the critical group.

Characterizes graphs that can be completely layer-stripped and bounds eigenvalue multiplicities.

Abstract

We propose an algebraic framework for generalized graph Laplacians which unifies the study of resistor networks, the critical group, and the eigenvalues of the Laplacian and adjacency matrices. Given a graph with boundary $G$ together with a generalized Laplacian $L$ with entries in a commutative ring $R$, we define a generalized critical group $\Upsilon_R(G,L)$. We relate $\Upsilon_R(G,L)$ to spaces of harmonic functions on the network using the Hom, Tor, and Ext functors of homological algebra. We study how these algebraic objects transform under combinatorial operations on the network $(G,L)$, including harmonic morphisms, layer-stripping, duality, and symmetry. In particular, we use layer-stripping operations from the theory of resistor networks to systematize discrete harmonic continuation. This leads to an algebraic characterization of the graphs with boundary that can be completely layer-stripped, an algorithm for simplifying computation of $\Upsilon_R(G,L)$, and upper bounds for the number of invariant factors in the critical group and the multiplicity of Laplacian eigenvalues in terms of geometric quantities.