Analytical topological bounds and favorable elastic paths in discrete molecular networks

Abstract

The mechanical response of discrete networks is fundamentally governed by their underlying connectivity. While traditional matrix-based approaches successfully capture global mechanical responses, their utility is delimited by an 𝒪(𝑁3)computational cost that limits direct access to specific topological pathways and localized load transmission. Here, we formulate a graph-theoretical framework that determines the effective elastic constant of molecular networks through analytical topological rules, bypassing global matrix diagonalization. Central to this formulation is the concept of favorable elastic paths—the specific topological highways that actively transmit mechanical force between terminal vertices. For series-parallel architectures, we compute effective stiffness via recursive edge reductions characterized by excluded minors. To quantify localized mechanical contributions, we introduce a normalized Relevance Ratio and a hierarchical (𝑘, 𝑑)-subgraph classification based on graph distance and vertex connectivity. For complex, non-reducible topologies, we leverage Menger's theorem to derive rigorous topological upper and lower bounds for the effective stiffness using purely combinatorial opera tions. By mapping atoms and bonds to vertices and edges, we demonstrate how discrete connectivity dictates continuum mechanical response. This topology-driven approach provides a computationally efficient (𝒪(∣ 𝐸 ∣)) and analytically transparent baseline, opening new avenues for the algorithmic screening and targeted chemical functionalization of complex biopolymer networks and mechanical metamaterials.