  Consider a weighted and undirected graph, possibly with self-loops, and its
corresponding Laplacian matrix, possibly augmented with additional diagonal
elements corresponding to the self-loops. The Kron reduction of this graph is
again a graph whose Laplacian matrix is obtained by the Schur complement of the
original Laplacian matrix with respect to a subset of nodes. The Kron reduction
process is ubiquitous in classic circuit theory and in related disciplines such
as electrical impedance tomography, smart grid monitoring, transient stability
assessment in power networks, or analysis and simulation of induction motors
and power electronics. More general applications of Kron reduction occur in
sparse matrix algorithms, multi-grid solvers, finite--element analysis, and
Markov chains. The Schur complement of a Laplacian matrix and related concepts
have also been studied under different names and as purely theoretic problems
in the literature on linear algebra. In this paper we propose a general
graph-theoretic framework for Kron reduction that leads to novel and deep
insights both on the mathematical and the physical side. We show the
applicability of our framework to various practical problem setups arising in
engineering applications and computation. Furthermore, we provide a
comprehensive and detailed graph-theoretic analysis of the Kron reduction
process encompassing topological, algebraic, spectral, resistive, and
sensitivity analyses. Throughout our theoretic elaborations we especially
emphasize the practical applicability of our results.
