On first and second stress polynomials of graphs

Shimbel (1953) introduced the node centrality index, which is the stress of a vertex. The number of geodesics (shortest paths) that pass through a vertex in a graph is its stress. A number that links a chemical structure to physical characteristics or chemical reactivity is called a topological index of the chemical structure (graph). This paper presents the first and second stress polynomials, which are new stress polynomials for graphs based on vertices stresses. Additionally, we compute stress polynomials for a few standard graphs, prove a few results, and establish a few inequalities.