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We propose an algebraic framework for generalized graph Laplacians whichunifies the study of resistor networks, the critical group, and the eigenvaluesof 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 operationson the network $(G,L)$, including harmonic morphisms, layer-stripping, duality,and symmetry. In particular, we use layer-stripping operations from the theoryof resistor networks to systematize discrete harmonic continuation. This leadsto an algebraic characterization of the graphs with boundary that can becompletely layer-stripped, an algorithm for simplifying computation of$\Upsilon_R(G,L)$, and upper bounds for the number of invariant factors in thecritical group and the multiplicity of Laplacian eigenvalues in terms ofgeometric quantities.

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