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The Generalized Euler Characteristics of the Graphs Split at Vertices.

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We discovered a relationship between generalized Euler characteristics of graphs and their subgraphs. This connection helps determine the number of vertices where split graphs were initially connected, verified with microwave network simulations.

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Area of Science:

  • Graph theory
  • Quantum chaos
  • Network analysis

Background:

  • Generalized Euler characteristics are topological invariants.
  • Splitting graphs can alter their topological properties.
  • Understanding connectivity in complex networks is crucial.

Purpose of the Study:

  • To establish a relationship between the generalized Euler characteristic of a graph and its disconnected subgraphs.
  • To determine the number of initial connection vertices using Euler characteristics.
  • To experimentally validate theoretical findings in a physical system.

Main Methods:

  • Theoretical derivation of the relationship between graph and subgraph generalized Euler characteristics.
  • Experimental simulation using microwave networks representing quantum graphs.
  • Analysis of vertices with Dirichlet boundary conditions (|VDo| and |VDi|).

Main Results:

  • A direct relationship was found between Eo(|VDo|) and Ei(|VDi|).
  • The generalized Euler characteristics accurately predict the number of initial connection vertices.
  • Experimental results using microwave networks confirmed the theoretical predictions.

Conclusions:

  • The generalized Euler characteristic provides a powerful tool for analyzing graph connectivity.
  • This method offers a way to determine the original connectivity of split networks.
  • The study bridges theoretical graph theory with experimental physics.