The Generalized Euler Characteristics of the Graphs Split at Vertices

Abstract

We show that there is a relationship between the generalized Euler characteristic Eo(|VDo|) of the original graph that was split at vertices into two disconnected subgraphs i=1,2 and their generalized Euler characteristics Ei(|VDi|). Here, |VDo| and |VDi| denote the numbers of vertices with the Dirichlet boundary conditions in the graphs. The theoretical results are experimentally verified using microwave networks that simulate quantum graphs. We demonstrate that the evaluation of the generalized Euler characteristics Eo(|VDo|) and Ei(|VDi|) allow us to determine the number of vertices where the two subgraphs were initially connected.

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

“…In this article we consider a general situation when an original graph . Simpler situations of quantum graphs split only at vertices or at edges were considered in [66,67] and [68], respectively.…”

Dissipative, Entropy Production Systems across Condensed Matter and Interdisciplinary Classical vs. Quantum Physics

“…The last in-presentation-order (QU) paper of the SI “The Generalized Euler Characteristics of the Graphs Split at Vertices” [ 10 ] argues that there is a relationship between the generalized Euler characteristic of an original graph that was split at vertices into two disconnected subgraphs, and their generalized Euler characteristics. The theoretical results are experimentally justified by employing microwave networks that mimic quantum graphs.…”

“…To summarize, this SI has focused on relevant and fundamental issues of statistical classical/quantum physics (and related subdisciplines), pointing to maximum-entropy and entropy production (and/or the spread of information) principles experienced by the respective CL and QU systems in (non)equilibrium conditions. The studies [ 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 ] disclose both the theoretical depth as well as the practical usefulness of the applied CL and QU approaches.…”