Algebraic graph theory is a cornerstone in the study of electrical networks ranging from miniature integrated circuits to continental-scale power systems. Conversely, many fundamental results of algebraic graph theory were laid out by early electrical circuit analysts. In this paper, we survey some fundamental and historic as well as recent results on how algebraic graph theory informs electrical network analysis, dynamics, and design. In particular, we review the algebraic and spectral properties of graph adjacency, Laplacian, incidence, and resistance matrices and how they relate to the analysis, network reduction, and dynamics of certain classes of electrical networks. We study these relations for models of increasing complexity ranging from static resistive direct current (dc) circuits, over dynamic resistor..inductor..capacitor (RLC) circuits, to nonlinear alternating current (ac) power flow. We conclude this paper by presenting a set of fundamental open questions at the intersection of algebraic graph theory and electrical networks.

We consider a graph-theoretic approach to the performance and robustness of a platoon of vehicles, in which each vehicle communicates with its k-nearest neighbors. In particular, we quantify the platoon's stability margin, robustness to disturbances (in terms of system H∞ norm), and maximum delay tolerance via graph-theoretic notions, such as nodal degrees and (grounded) Laplacian matrix eigenvalues. The results show that there is a trade-off between robustness to time delay and robustness to disturbances. Both lurst-order dynamics (reference velocity tracking) and second-order dynamics (controlling inter-vehicular distance) are analyzed in this direction. Theoretical contributions are conlurmed via simulation results.