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Simplex polynomial in complex networks and its applications to compute the Euler characteristic.
Zhaoyang Wang1,2, Xianghui Fu2,3, Bo Deng2,3
1School of Computer, Qinghai Normal University, Xining, China.
Frontiers in Computational Neuroscience
|December 12, 2025
Summary
This study introduces the simplex polynomial, a novel tool for calculating the Euler characteristic. This new method provides a deeper understanding of topological invariants and their applications in network structures.
Area of Science:
- Algebraic topology
- Graph theory
- Network science
Background:
- The Euler characteristic is a topological invariant with broad applications.
- Current methods for calculation include simplicial decomposition and the Euler-Poincaré formula.
- Simplicial complexes are fundamental structures in algebraic topology.
Purpose of the Study:
- Introduce a new polynomial, the simplex polynomial, for analyzing simplicial complexes.
- Develop a novel method for computing the Euler characteristic using the simplex polynomial.
- Investigate the properties of the simplex polynomial and its relation to network structures.
Main Methods:
- Definition and exploration of the properties of the simplex polynomial.
- Application of the simplex polynomial to compute the Euler characteristic.
- Construction of simplicial complex structures to prove the existence of the Euler characteristic.
- Analysis of recurrence relations for simplex polynomials in common network structures.
Main Results:
- A new method for computing the Euler characteristic is presented.
- The existence of the Euler characteristic as an arbitrary integer is proven through simplicial complex construction.
- A class of simplicial complex structures corresponding to an Euler characteristic of 1 is identified.
- Recurrence relations for simplex polynomials and Euler characteristics of three common network structures are derived.
Conclusions:
- The simplex polynomial offers a new perspective on calculating topological invariants.
- The study establishes a link between graph theory (simplex polynomial) and algebraic topology (Euler characteristic).
- Further research is suggested through three open questions for the interested readers.
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