Simplex polynomial in complex networks and its applications to compute Euler characteristic

Zhaoyang Wang¹Xianghui Fu¹Bo Deng¹Yang Chen^2*Haixing Zhao^3*

• ¹Qinghai Normal University, Xining, China
• ²Shandong Technology and Business University, Yantai, China
• ³Qinghai Nationalities University, Xining, China

In algebraic topology, a k -dimensional simplex is defined as a convex polytope consisting of  k vertices. If spatial dimensionality is not considered, it corresponds to the complete graph with  k vertices in graph theory. The alternating sum of the number of simplexes across dimensions yields a topological invariant known as the Euler characteristic, which has gained significant attention due to its widespread application in fields such as topology, homology theory, complex systems, and biology. The most common method for calculating the Euler characteristic is through simplicial decomposition and the Euler-Poincaré formula. In this paper, we introduce a new "subgraph" polynomial, termed the simplex polynomial, and explore some of its properties. Using those properties, we provide a new method for computing the Euler characteristic and prove the existence of the Euler characteristic as an arbitrary integer by constructing the corresponding simplicial complex structure. When Euler characteristic is 1, we determined a class of corresponding simplicial complex structure. Moreover, for three common network structures, we present the recurrence relations for their simplex polynomials and their corresponding Euler characteristics. Finally, at the end of this work, three basic questions are raised for the interested readers to study deeply.