Investigation of the generalized Euler characteristic of graphs and microwave networks split at edges and vertices

Abstract

We analyze the situation when the original graph is split at edges and vertices into two disconnected subgraphs. We show that there is a relationship between the generalized Euler characteristic Eo(|VDo|) of the original graph and the generalized Euler characteristics Ei(|VDi|), i = 1, 2, of two disconnected subgraphs, where |VDo| and |VDi|, i = 1, 2, are the numbers of vertices with the Dirichlet boundary conditions in the graphs. Theoretical predictions are verified experimentally using microwave networks which simulate quantum graphs.

Toward Remote Physical-Model-Based Fault Localization in Transmission-Line Networks

“…Meanwhile, we note that other physics-based remote-sensing approaches for transmission-line networks are currently being explored in the literature. For example, [3] considers the case of an initial network being split at various edges and nodes into two networks; it is shown in [3] that by determining the networks' Euler characteristic from scattering measurements [4], one can determine at how many edges and nodes the initial network was spliteven without knowing its topology. Related to the present paper is also a "differential DORT" technique † for the detection of soft faults in complex wire networks that is similarly based on measuring the scattering matrix of the intact and faulty network, as well as knowledge of the network topology [5].…”

Selected Properties of Microwave Networks with Orthogonal and Symplectic Symmetries

“…Here, we discuss the situation when the original graph with orthogonal symmetry is split at edges and vertices into two disconnected subgraphs [2][3]. We show that there is a relationship between the generalized Euler characteristic of the original graph and the generalized Euler characteristics of two disconnected subgraphs.…”