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

In algebraic topology, a <i>k</i>-dimensional simplex is defined as a convex polytope consisting of <i>k</i> + 1 vertices. If spatial dimensionality is not considered, it corresponds to the complete graph with <i>k</i> + 1 vertices in graph theory. The alternating sum of the number of simplices 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 study, 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 the Euler characteristic is 1, we determined a class of corresponding simplicial complex structures. 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 study, three basic questions are raised for the interested readers to study deeply.